Global $W^{2,p}$ Regularity in Optimal Transport

Shibing Chen Address: School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China Email address: chenshib@ustc.edu.cn , Yuanyuan Li Address: Institute for Theoretical Sciences, Westlake University, Hangzhou, 310030, China Email address: lyyuan@westlake.edu.cn and Xianduo Wang Address: School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, China Email address: wangxd@hust.edu.cn

Abstract.

In this paper we establish global $W^{2,p}$ estimates for the convex potentials of quadratic optimal transport between bounded convex domains with continuous positive densities. All the assumptions are optimal. The main new ideas include a blow-up analysis that allows for lower-dimensional collapse of the limiting source measure and reduces the limiting problem to a transport problem on its affine hull, and a good–bad scale decomposition in which rigidity controls the good scales while a counting argument shows that the proportion of bad scales tends to zero.

1. Introduction

Let $\Omega,\Omega^{*}\subset\mathbb{R}^{n}$ be bounded open convex sets, and assume that

(1.1)

f\in C(\overline{\Omega}),\qquad g\in C(\overline{\Omega^{*}}),\qquad 0<\lambda\leq f,g\leq\Lambda,\qquad\int_{\Omega}f=\int_{\Omega^{*}}g.

Let $u$ and $v$ be the dual convex potentials for the quadratic optimal transport, so that

(1.2)

(Du)_{\#}(f\,\,\mathrm{d}x)=g\,\,\mathrm{d}y,\qquad(Dv)_{\#}(g\,\,\mathrm{d}y)=f\,\,\mathrm{d}x.

As usual, the potentials are extended to $\mathbb{R}^{n}$ by the minimal convex extension described in Section 2.

The existence and uniqueness of the optimal transport map were established by Brenier [1]; see also McCann [21] for the more general theory of monotone maps. For the interior regularity theory of the Monge–Ampère equation, we refer the reader to the fundamental works [2, 3, 4, 14, 15]. For general background on optimal transport and the Monge–Ampère equation, we refer to the monographs [27, 28, 16]. The fundamental regularity condition for optimal transport with general cost functions was introduced by Ma–Trudinger–Wang [20], and was subsequently shown by Loeper [19] to be necessary for the continuity of optimal transport maps for arbitrary smooth positive densities. For global Sobolev regularity of optimal transport potentials on a class of nonconvex planar domains, see [22, 11].

Under stronger smoothness assumptions, Delanoë [13] first established global classical solvability in dimension two for uniformly convex domains. Caffarelli [5] subsequently established global $C^{1,\alpha}$ estimates for bounded convex domains when the densities are bounded above and below. Later, in [6], he proved global $C^{1,1-\varepsilon}$ estimates when the domains are uniformly convex and $C^{2}$ and the densities are positive and continuous. In the same paper, Caffarelli also established global $C^{2,\alpha^{\prime}}$ estimates, assuming further that the densities are $C^{\alpha}$ , for some $\alpha^{\prime}<\alpha$ . Urbas [26] extended the classical solvability result of Delanoë to higher dimensions.

Chen–Liu–Wang [8] proved global $C^{1,1-\varepsilon}$ and $W^{2,p}$ estimates for continuous positive densities when the convex domains have $C^{1,1}$ boundaries, and established $C^{2,\alpha}$ estimates when the domains are $C^{1,1}$ and the densities are $C^{\alpha}$ . Savin–Yu obtained global $C^{1,1-\varepsilon}$ and $W^{2,p}$ estimates on arbitrary planar convex domains with constant densities [25]. By introducing a very useful monotonicity formula, a related version of which was also used by Yuan [29] to give a new proof of the Pogorelov estimate for the Monge–Ampère equation, Collins–Tong extended the results of Savin–Yu to all dimensions under a Hölder continuity assumption on the densities [12]. They also established global $C^{2,\alpha}$ estimates under the sharp assumptions, namely, that the domains are convex and $C^{1,\alpha}$ and that the densities are $C^{\alpha}$ . However, $W^{2,p}$ estimates under the sharp assumptions have remained open. In this paper, we give a complete answer.

Theorem 1.1.

Suppose (1.1) and (1.2) hold. Then, for every $p>1$ ,

u\in W^{2,p}(\overline{\Omega}),\qquad v\in W^{2,p}(\overline{\Omega^{*}}).

There are three main difficulties. First, near the boundary, sections may be cut by the domains in an irregular way, since the domains are not assumed to be smooth or strictly convex. Second, continuity of the densities only implies that their oscillations tend to zero, which does not imply that the errors over all scales are summable. Third, after affine normalization, the limiting source measure may be supported on a lower dimensional subspace.

The proof does not require summability of the density errors: their average over dyadic scales tends to zero, so only $o(N)$ scales are bad. The blow-up analysis also allows the limiting source measure to collapse onto a proper subspace.

The proof consists of the following five steps:

(i)

An almost-monotonicity estimate. Fix $x_{0}\in\overline{\Omega}$ , set $y_{0}=Du(x_{0})$ , and define

D_{r}(u,x_{0}):=\bigl\{x\in\Omega:(x-x_{0})\cdot(Du(x)-y_{0})<r^{2}\bigr\},\qquad F_{x_{0}}(r):=r^{-n}\int_{D_{r}(u,x_{0})}f(x)\,\,\mathrm{d}x.

Adapting the monotonicity formula and first-variation argument of Collins–Tong [12, Theorem 3.1 and Proposition 3.1], we prove that

F_{x_{0}}(r_{1})\leq F_{x_{0}}(r_{2})\exp\!\left(C\int_{r_{2}}^{r_{1}}\frac{\omega_{f}(Cs^{\sigma})+\omega_{g}(Cs^{\sigma})}{s}\,\,\mathrm{d}s\right)

whenever $0<r_{2}<r_{1}\leq r_{0}$ . The error on the right is additive on adjacent scale intervals. Although it need not be summable down to zero, its average over dyadic logarithmic scales tends to zero. Note that no Dini condition is required.

(ii)

Blow-up analysis. We perform the normalization in the same spirit as in [9]. More precisely, the normalization is applied to sections of the source potential, in contrast to [9], where it is applied only on the target side. We normalize centered sections using their John ellipsoids. The geometric decay of the centered sections controls the normalized potentials and yields a locally uniform limit that is finite on all of $\mathbb{R}^{n}$ . At the same time, we consider the limits of the corresponding dual potentials. This allows us to handle the possible degeneration in which the limiting source measure is supported on a proper linear subspace.
(iii)

The limit and dimensional reduction. In the full-dimensional case, we follow [12, Theorem 4.1]. If the limiting source measure is supported on an $m$ -dimensional subspace $L$ , with $q=n-m$ , projection onto $L$ identifies the limiting density with the normalized $q$ -dimensional volumes of transverse sections. By convexity and the Brunn–Minkowski inequality, its $q$ -th root is concave, and the projected measures have uniformly bounded densities. The same argument applies to the projected target measure. The mass identity then yields a two-homogeneous $m$ -dimensional limiting transport problem, which can be lifted back to the original variables.
(iv)

Rigidity and iteration. The rigidity of the blow-up limits implies that, on a sufficiently long interval of scales with small total error, the next sub-level set differs little from one half of the preceding one. The sum of the errors over the first $N$ dyadic scales is $o(N)$ , so the estimate fails at only $o(N)$ indices. Consequently, for every $\delta>0$ ,

$B_{c_{\delta}h^{1/2+\delta}}(x)\subset S_{h}^{c}[u](x)\subset B_{C_{\delta}h^{1/2-\delta}}(x).$
(v)

Conclusion. Finally, the $W^{2,p}$ estimate follows from combining Caffarelli’s interior estimate with a covering argument of Savin [24].

We would also like to point out that the methods developed in this paper can be used to slightly improve the results in [10]. More precisely, one can relax the regularity assumptions on the target domains by removing the uniform convexity condition and requiring only $C^{1,\alpha}$ regularity (resp. mere convexity) to obtain $C^{2,\alpha}$ (resp. $C^{1,1-\epsilon}$ ) regularity of the singular set.

The paper is organized as follows. Section 2 contains the notation and the standard estimates for sub-level sets. In Section 3 we prove the mass estimate. Sections 4 and 5 contain the blow-up analysis and dimension reduction arguments. The limit profile is identified in Section 6, and the iteration and the proof of the main theorem are carried out in Section 7. Appendix A contains the local regularity result used for locally finite measures.

2. Preliminaries

Throughout the paper, $c,C>0$ denote universal constants depending only on the fixed data. Note that their values may change from line to line.

Let $u$ and $v$ be the potentials satisfying (1.2). For a set $D$ and a convex function $\phi$ , define

I_{D}(z)=\begin{cases}0,&z\in D,\\ +\infty,&z\notin D,\end{cases}\qquad\phi^{*}(z)=\sup_{\xi\in\mathbb{R}^{n}}\{z\cdot\xi-\phi(\xi)\}.

Then we extend $u,v$ to convex functions on $\mathbb{R}^{n}$ by

(2.1)

\overline{u}=(v+I_{\overline{\Omega^{*}}})^{*},\qquad\overline{v}=(u+I_{\overline{\Omega}})^{*}.

We call $\overline{u},\overline{v}$ the minimal convex extensions of $u,v$ . They are finite on $\mathbb{R}^{n}$ , agree with $u,v$ on their respective closed domains, and satisfy

(2.2)

\overline{u}^{*}=v+I_{\overline{\Omega^{*}}},\qquad\overline{v}^{*}=u+I_{\overline{\Omega}}.

Note that $(x,y)\in\overline{\Omega}\times\overline{\Omega^{*}}$ satisfies

u(x)+v(y)=x\cdot y

if and only if $y\in\partial\overline{u}(x)$ and $x\in\partial\overline{v}(y)$ .

For $x_{0}\in\overline{\Omega}$ , let

\ell_{x_{0}}(x)=u(x_{0})+Du(x_{0})\cdot(x-x_{0}).

For $h>0$ , we define

S_{h}[u](x_{0})=\{x\in\Omega:u(x)<\ell_{x_{0}}(x)+h\}.

The centered sub-level set of height $h$ at $x_{0}$ is

(2.3)

S_{h}^{c}[u](x_{0})=\{x\in\mathbb{R}^{n}:\overline{u}(x)<\widehat{\ell}(x)+h\},

where $\widehat{\ell}$ is affine, $\widehat{\ell}(x_{0})=u(x_{0})$ , and is chosen so that $x_{0}$ is the center of mass of $S_{h}^{c}[u](x_{0})$ . The existence of $\widehat{\ell}$ follows from [6, Section 2]. We also set

D_{r}(u,x_{0})=\bigl\{x\in\Omega:(x-x_{0})\cdot\bigl(Du(x)-Du(x_{0})\bigr)<r^{2}\bigr\},\qquad r>0.

Note that $S_{h}[u](x_{0})$ and $D_{r}(u,x_{0})$ are contained in $\Omega,$ but $S_{h}^{c}[u](x_{0})$ may contain both points in and out of $\Omega$ . When no confusion arises, we may abbreviate them as $S_{h}[u],S_{h}^{c}[u]$ or $S_{h}(x_{0}),S_{h}^{c}(x_{0})$ . The corresponding sets for $v$ are defined by interchanging $(\Omega,u)$ and $(\Omega^{*},v)$ .

We first recall the estimates used throughout the paper.

Proposition 2.1.

There are $\alpha_{0}\in(0,1)$ and constants depending only on $n,\lambda,\Lambda$ and the fixed inner and outer radii of the two domains such that the following hold.

(i)

By [18, Theorem 1.1] and [12, Proposition 2.5], we have

$[Du]_{C^{0,\alpha_{0}}(\overline{\Omega})}+[Dv]_{C^{0,\alpha_{0}}(\overline{\Omega^{*}})}\leq C.$

For $x_{0}\in\overline{\Omega}$ , $y_{0}\in\overline{\Omega^{*}}$ , and every fixed $R<\infty$ ,

(2.4) $[D\overline{u}]_{C^{0,\alpha_{0}}(B_{R}(x_{0}))}+[D\overline{v}]_{C^{0,\alpha_{0}}(B_{R}(y_{0}))}\leq C_{R}.$
(ii)

By [5], we know the restrictions

$Du:\Omega\longrightarrow\Omega^{*},\qquad Dv:\Omega^{*}\longrightarrow\Omega$

are mutually inverse homeomorphisms.

(iii)

By [12, Proposition 2.6] and [12, Proposition 2.3(3)], for every $x_{0}\in\overline{\Omega}$ , $r>0$ , and $h>0$ ,

(2.5)		$\displaystyle x_{0}+\tfrac{1}{2}\bigl(S_{r^{2}}[u](x_{0})-x_{0}\bigr)$	$\displaystyle\subset D_{r}(u,x_{0})\subset S_{r^{2}}[u](x_{0}),$
(2.6)		$\displaystyle S_{ch}^{c}[u](x_{0})\cap\Omega$	$\displaystyle\subset S_{h}[u](x_{0})\subset S_{Ch}^{c}[u](x_{0})\cap\Omega.$

The analogous inclusions hold for $v$ .

(iv)

By [12, Proposition 2.3(1),(2)], let $p_{h}=D\widehat{\ell}$ , where $\widehat{\ell}$ defines $S_{h}^{c}[u](x_{0})$ in (2.3), and let $A_{h}$ be the positive definite linear map which normalizes the John ellipsoid of $S_{h}^{c}[u](x_{0})-x_{0}$ onto $B_{1}$ . Then

(2.7)		$\displaystyle B_{c}$	$\displaystyle\subset h^{-1}A_{h}^{-t}\bigl(Du(S_{h}^{c}[u](x_{0}))-p_{h}\bigr)\subset B_{C},$
(2.8)		$\displaystyle ch^{n}$	$\displaystyle\leq\|S_{h}^{c}[u](x_{0})\|\,\bigl\|Du(S_{h}^{c}[u](x_{0}))\bigr\|\leq Ch^{n}.$

The same estimates hold with $u$ replaced by $v$ .

(v)

For every $0<h_{-}\leq h_{+}<\infty$ , there are $0<c_{h_{-},h_{+}}\leq C_{h_{-},h_{+}}<\infty$ such that, for every $x_{0}\in\overline{\Omega}$ , $y_{0}=D\overline{u}(x_{0})$ , and $h\in[h_{-},h_{+}]$ ,

	$\displaystyle B_{c_{h_{-},h_{+}}}(x_{0})$	$\displaystyle\subset S_{h}^{c}[u](x_{0})\subset B_{C_{h_{-},h_{+}}}(x_{0}),$
	$\displaystyle B_{c_{h_{-},h_{+}}}(y_{0})$	$\displaystyle\subset S_{h}^{c}[v](y_{0})\subset B_{C_{h_{-},h_{+}}}(y_{0}).$

Proof.

We only need to prove (v) and it suffices to give the proof for $u$ . Suppose that the outer inclusion fails. Then there are $x_{k}\in\overline{\Omega}$ , $h_{k}\in[h_{-},h_{+}]$ , and centered sections $S_{h_{k}}^{c}[u](x_{k})$ whose outer radii tend to infinity. After passing to a subsequence,

x_{k}\to x_{\infty},\qquad D\overline{u}(x_{k})\to D\overline{u}(x_{\infty}),\qquad h_{k}\to h_{\infty},

and the convex functions

z\longmapsto\overline{u}(x_{k}+z)-\overline{u}(x_{k})-D\overline{u}(x_{k})\cdot z

converge locally uniformly to a convex function $\phi_{\infty}$ . Let $e_{k}$ be a direction of maximal radius and assume that $e_{k}\to e$ . Both radial endpoints on $x_{k}+\mathbb{R}e_{k}$ tend to infinity. Thus, for every fixed $t>0$ , the points $x_{k}\pm te_{k}$ belong to $S_{h_{k}}^{c}$ for all large $k$ . Adding the two defining inequalities and passing to the limit, we obtain

0\leq\phi_{\infty}(te)+\phi_{\infty}(-te)\leq 2h_{+}.

The function $\phi_{\infty}(te)+\phi_{\infty}(-te)$ is convex in $t$ , vanishes at $0$ , and is bounded on $[0,\infty)$ . Therefore, it is identically zero. Thus $u$ is affine on the line $x_{\infty}+\mathbb{R}e$ , which contradicts the fact that its subgradient image has nonempty interior. This proves the uniform outer inclusion.

For the inner inclusion, let $p_{h}=D\widehat{\ell}_{h}$ . The existence of a bounded section of $\overline{u}-p_{h}\cdot(\,\cdot-x_{0})$ implies

p_{h}\in\operatorname{dom}\overline{u}^{*}=\overline{\Omega^{*}}.

Set $L_{0}=1+\sup_{y\in\overline{\Omega^{*}}}|y|$ . Since $D\overline{u}(\mathbb{R}^{n})\subset\overline{\Omega^{*}}$ , both $\overline{u}$ and the affine function with slope $p_{h}$ are $L_{0}$ -Lipschitz. Therefore,

\overline{u}(x_{0}+z)-\overline{u}(x_{0})-p_{h}\cdot z\leq 2L_{0}|z|<h_{-}\leq h

whenever $|z|<h_{-}/(2L_{0})$ and the argument for $v$ is the same. ∎

We shall also use the following two elementary consequences.

Lemma 2.2.

Let $\phi:\mathbb{R}^{n}\to\mathbb{R}$ be finite and convex, with $\phi(0)=0$ and $0\in\partial\phi(0)$ . If

B_{r}\subset S_{h}^{c}[\phi](0)=\{x\in\mathbb{R}^{n}:\phi(x)-p_{h}\cdot x<h\},

then $|p_{h}|\leq h/r$ .

Proof.

Since $0\in\partial\phi(0)$ and $\phi(0)=0$ , we have $\phi\geq 0$ . If $p_{h}\neq 0$ , set $e=p_{h}/|p_{h}|$ . For $0<\varepsilon<r$ , the point $-(r-\varepsilon)e$ belongs to $S_{h}^{c}[\phi](0)$ , and hence

0\leq\phi(-(r-\varepsilon)e)<h-(r-\varepsilon)|p_{h}|.

Letting $\varepsilon\rightarrow 0$ proves the assertion. The case $p_{h}=0$ is immediate. ∎

The next lemma is standard and it compares centered sections at two heights.

Lemma 2.3.

Let $\phi:\mathbb{R}^{n}\to\mathbb{R}$ be finite and convex, and suppose that $S_{h}^{c}[\phi](0)$ is bounded. Then, for every $0<\theta\leq 1$ ,

(2.9)

\frac{\theta}{n^{3}}S_{h}^{c}[\phi](0)\subset S_{\theta h}^{c}[\phi](0)\subset n^{3}S_{h}^{c}[\phi](0).

Proof.

After subtracting an affine function, $\phi(0)=0$ and $\phi\geq 0$ . Fix $e\in\mathbb{S}^{n-1}$ , and write the two boundary points of $S_{h}^{c}\cap\mathbb{R}e$ as $a_{h}e$ and $-b_{h}e$ , where $a_{h},b_{h}>0$ . Set $R_{h}=\max\{a_{h},b_{h}\}$ . Since $S_{h}^{c}$ is centered at the origin, by John’s lemma [7, Lemma A.3], with $\gamma_{n}=n^{3/2}$ , we have

(2.10)

\frac{R_{h}}{\gamma_{n}}\leq a_{h},b_{h}\leq R_{h}.

Since $\frac{\phi(a_{h}e)}{a_{h}}+\frac{\phi(-b_{h}e)}{b_{h}}=h\left(\frac{1}{a_{h}}+\frac{1}{b_{h}}\right),$ and $t\mapsto\phi(\pm te)/t$ is nondecreasing on $(0,\infty)$ , the even convex function $G_{e}(t)=\phi(te)+\phi(-te)$ satisfies

(2.11)

G_{e}(R_{h})\geq 2h,\qquad G_{e}(R_{h}/\gamma_{n})\leq 2h.

Let $R_{\theta h}$ be defined in the same way at height $\theta h$ . If $R_{\theta h}<R_{h}/\gamma_{n}$ , the monotonicity of $G_{e}(t)/t$ and (2.11) imply

\frac{2\theta h}{R_{\theta h}}\leq\frac{2\gamma_{n}h}{R_{h}}.

Thus $R_{\theta h}\geq\theta R_{h}/\gamma_{n}$ . The same conclusion is immediate when $R_{\theta h}\geq R_{h}/\gamma_{n}$ . If $0<\theta<1$ , then

G_{e}(R_{\theta h}/\gamma_{n})\leq 2\theta h<2h\leq G_{e}(R_{h}),

so $R_{\theta h}\leq\gamma_{n}R_{h}$ . This inequality is automatic when $\theta=1$ . Combining these bounds with (2.10) at the two heights, we obtain

\frac{\theta}{\gamma_{n}^{2}}a_{h}\leq a_{\theta h}\leq\gamma_{n}^{2}a_{h}

in every direction. Since $\gamma_{n}^{2}=n^{3}$ , (2.9) follows. ∎

3. An almost monotonicity estimate

Fix $x_{0}\in\overline{\Omega}$ and set $y_{0}=D\overline{u}(x_{0})$ . Define

\Psi_{x_{0}}(x):=(x-x_{0})\cdot\bigl(Du(x)-y_{0}\bigr),\qquad\Psi_{y_{0}}^{*}(y):=(y-y_{0})\cdot\bigl(Dv(y)-x_{0}\bigr),

and, for $h>0$ ,

D_{\sqrt{h}}(u,x_{0})=\{x\in\Omega:\Psi_{x_{0}}(x)<h\},\qquad D_{\sqrt{h}}(v,y_{0})=\{y\in\Omega^{*}:\Psi_{y_{0}}^{*}(y)<h\}.

When $y=Du(x)$ , the Legendre identities imply $\Psi_{y_{0}}^{*}(y)=\Psi_{x_{0}}(x)$ . Hence

(3.1)

M_{x_{0}}(h):=\int_{D_{\sqrt{h}}(u,x_{0})}f(x)\,\,\mathrm{d}x=\int_{D_{\sqrt{h}}(v,y_{0})}g(y)\,\,\mathrm{d}y.

Define the two global moduli

	$\displaystyle\omega_{f}(t)$	$\displaystyle=\sup\{\|f(x)-f(x^{\prime})\|:x,x^{\prime}\in\overline{\Omega},\ \|x-x^{\prime}\|\leq t\},$
	$\displaystyle\omega_{g}(t)$	$\displaystyle=\sup\{\|g(y)-g(y^{\prime})\|:y,y^{\prime}\in\overline{\Omega^{*}},\ \|y-y^{\prime}\|\leq t\}.$

By (2.4), for every fixed $R<\infty$ and $|z|,|\zeta|\leq R$ ,

	$\displaystyle 0$	$\displaystyle\leq\overline{u}(x_{0}+z)-\overline{u}(x_{0})-y_{0}\cdot z\leq C_{R}\|z\|^{1+\alpha_{0}},$
	$\displaystyle 0$	$\displaystyle\leq\overline{v}(y_{0}+\zeta)-\overline{v}(y_{0})-x_{0}\cdot\zeta\leq C_{R}\|\zeta\|^{1+\alpha_{0}}.$

It follows from (2.1) and the above estimates that

	$\displaystyle u(x)-u(x_{0})-y_{0}\cdot(x-x_{0})$	$\displaystyle\geq c\|x-x_{0}\|^{1+1/\alpha_{0}},$
	$\displaystyle v(y)-v(y_{0})-x_{0}\cdot(y-y_{0})$	$\displaystyle\geq c\|y-y_{0}\|^{1+1/\alpha_{0}},$

for $x\in\overline{\Omega}$ , $y\in\overline{\Omega^{*}}$ , uniformly in $x_{0},y_{0}$ .

If $y=Du(x)$ , a direct computation shows that

\Psi_{x_{0}}(x)=u(x)-u(x_{0})-y_{0}\cdot(x-x_{0})+v(y)-v(y_{0})-x_{0}\cdot(y-y_{0}).

Both terms on the right are nonnegative. Consequently, there are $C_{0},r_{0}>0$ , depending only on the fixed data, such that

(3.2)

\operatorname{diam}D_{r}(u,x_{0})+\operatorname{diam}D_{r}(v,y_{0})\leq C_{0}r^{\sigma},\qquad\sigma=\frac{2\alpha_{0}}{1+\alpha_{0}},\qquad 0<r\leq r_{0}.

Define

\vartheta(r)=\omega_{f}(C_{0}r^{\sigma})+\omega_{g}(C_{0}r^{\sigma}),\qquad F_{x_{0}}(r)=r^{-n}M_{x_{0}}(r^{2}).

For homogeneous densities, Collins–Tong established an exact monotonicity formula [12, Theorem 3.1]. Their smooth computation [12, Proposition 3.1] also identifies the error terms associated with nonhomogeneous Hölder densities. We give the argument because the densities are only continuous here and we estimate these terms directly by $\omega_{f}$ and $\omega_{g}$ .

Theorem 3.1.

There is a constant $C$ , depending only on $n,\lambda,\Lambda$ , such that

(3.3)

F_{x_{0}}(r_{1})\leq F_{x_{0}}(r_{2})\exp\!\left(C\int_{r_{2}}^{r_{1}}\frac{\vartheta(s)}{s}\,\,\mathrm{d}s\right)

whenever $0<r_{2}<r_{1}\leq r_{0}$ .

Proof.

After a change of coordinates and subtraction of an affine function, we may assume that $x_{0}=y_{0}=0$ and

u(0)=v(0)=0,\qquad Du(0)=Dv(0)=0,\qquad\Psi(x)=x\cdot Du(x),\qquad\Psi^{*}(y)=y\cdot Dv(y).

Step 1: the smooth case. Assume first that the domains are smooth and uniformly convex and that the data and potentials are smooth. These assumptions are used only in the coarea formula, the divergence theorem, and the classical change of variables. The resulting estimate is independent of all higher-order norms. Set $T=Du$ . A direct computation shows

D\Psi=T+D^{2}u(x)x,\qquad D\Psi^{*}(T(x))=x+\left(D^{2}u(x)\right)^{-1}T(x).

For almost every $h>0$ , we have

(3.4)

M^{\prime}(h)=\int_{\Omega\cap\{\Psi=h\}}\frac{f}{|D\Psi|}\,\,\mathrm{d}\mathcal{H}^{n-1},

where $M(h)=M_{x_{0}}(h)$ . Define

	$\displaystyle A_{f}(h)$	$\displaystyle=\int_{\Omega\cap\{\Psi=h\}}f\frac{x\cdot D\Psi}{\|D\Psi\|}\,\,\mathrm{d}\mathcal{H}^{n-1},$
	$\displaystyle I_{f}(h)$	$\displaystyle=\int_{\Omega\cap\{\Psi<h\}}x\cdot Df\,\,\mathrm{d}x,$
	$\displaystyle\mathcal{B}_{f}(h)$	$\displaystyle=\int_{\partial\Omega\cap\{\Psi<h\}}f\,x\cdot\nu_{\Omega}\,\,\mathrm{d}\mathcal{H}^{n-1},$

and define $A_{g},I_{g},\mathcal{B}_{g}$ analogously on the target. The divergence theorem gives

(3.5)

A_{f}+\mathcal{B}_{f}=nM+I_{f},\qquad A_{g}+\mathcal{B}_{g}=nM+I_{g}.

A direct computation shows that

d\mathcal{H}^{n-1}_{y}=\det D^{2}u\,\frac{|D\Psi^{*}(T(x))|}{|D\Psi(x)|}\,d\mathcal{H}^{n-1}_{x}.

Pulling the target coarea formula back by $T$ , as in [12, proof of Proposition 3.1], we obtain

(3.6)

A_{f}(h)+A_{g}(h)=4hM^{\prime}(h)+\mathcal{E}(h),

where

\mathcal{E}(h)=\int_{\Omega\cap\{\Psi=h\}}\frac{f}{|D\Psi|}\left|\left(D^{2}u(x)\right)^{1/2}x-\left(D^{2}u(x)\right)^{-1/2}T(x)\right|^{2}\,\,\mathrm{d}\mathcal{H}^{n-1}\geq 0.

Combining (3.5) and (3.6), we find

(3.7)

4hM^{\prime}(h)-2nM(h)=I_{f}(h)+I_{g}(h)-\mathcal{B}_{f}(h)-\mathcal{B}_{g}(h)-\mathcal{E}(h).

Since $0\in\overline{\Omega}\cap\overline{\Omega^{*}}$ , by convexity of the two domains, we have

x\cdot\nu_{\Omega}(x)\geq 0,\qquad y\cdot\nu_{\Omega^{*}}(y)\geq 0.

Consequently, $\mathcal{B}_{f},\mathcal{B}_{g}\geq 0$ .

It remains to estimate $I_{f}$ and $I_{g}$ . On every admissible ray $x=t\theta$ , the function

p_{\theta}(t)=\theta\cdot Du(t\theta)

is nonnegative and nondecreasing. Thus $D_{\sqrt{h}}(u,0)$ is star-shaped about the origin. If $R_{h}(\theta)$ is its radial endpoint, integration by parts on each ray gives

I_{f}(h)=n\int_{\mathbb{S}^{n-1}}\int_{0}^{R_{h}(\theta)}\bigl[f(R_{h}(\theta)\theta)-f(t\theta)\bigr]t^{n-1}\,\,\mathrm{d}t\,\,\mathrm{d}\theta.

Therefore

(3.8)		$\displaystyle(I_{f}(h))_{+}$	$\displaystyle\leq\frac{n}{\lambda}\omega_{f}\!\left(\operatorname{diam}D_{\sqrt{h}}(u,0)\right)M(h),$
(3.8)		$\displaystyle(I_{g}(h))_{+}$	$\displaystyle\leq\frac{n}{\lambda}\omega_{g}\!\left(\operatorname{diam}D_{\sqrt{h}}(v,0)\right)M(h).$

By (3.8), (3.7) and (3.2), we have

\frac{\,\mathrm{d}}{\,\mathrm{d}h}\bigl(h^{-n/2}M(h)\bigr)\leq C\frac{\vartheta(\sqrt{h})}{h}h^{-n/2}M(h)

for almost every $h\in(0,r_{0}^{2}]$ . Since $F_{x_{0}}(r)=r^{-n}M(r^{2})$ , it follows

F_{x_{0}}^{\prime}(r)\leq C\frac{\vartheta(r)}{r}F_{x_{0}}(r)\quad\text{for a.e. }r\in(0,r_{0}].

in the smooth setting. Integration proves (3.3).

Step 2: approximation. Let $P_{\overline{\Omega}},P_{\overline{\Omega^{*}}}$ be the metric projections onto the closed convex domains, namely,

P_{\overline{\Omega}}(x)=\operatorname*{argmin}_{z\in\overline{\Omega}}|x-z|,\ \ P_{\overline{\Omega^{*}}}(x)=\operatorname*{argmin}_{z\in\overline{\Omega^{*}}}|x-z|.

Choose smooth uniformly convex outer approximations

\Omega_{j}\rightarrow\overline{\Omega},\qquad\Omega_{j}^{*}\rightarrow\overline{\Omega^{*}},

and for $\varepsilon_{j}\rightarrow 0$ , define

f_{j}=(f\circ P_{\overline{\Omega}})*\rho_{\varepsilon_{j}},\qquad g_{j}=\kappa_{j}\bigl((g\circ P_{\overline{\Omega^{*}}})*\rho_{\varepsilon_{j}}\bigr),\qquad\kappa_{j}=\frac{\int_{\Omega_{j}}f_{j}\,\,\mathrm{d}x}{\int_{\Omega_{j}^{*}}\bigl((g\circ P_{\overline{\Omega^{*}}})*\rho_{\varepsilon_{j}}\bigr)\,\,\mathrm{d}y}.

The metric projections are one-Lipschitz, convolution does not increase a modulus of continuity, and the outer approximations converge in measure. Hence $\kappa_{j}\to 1$ , and, for all sufficiently large $j$ ,

(3.9)

\frac{\lambda}{2}\leq f_{j},g_{j}\leq 2\Lambda,\qquad\omega_{f_{j}}\leq\omega_{f},\qquad\omega_{g_{j}}\leq 2\omega_{g}.

Let $u_{j}$ be the corresponding transport potentials, extended to $\mathbb{R}^{n}$ as in Section 2. After the normalization, the approximating domains have uniform inner and outer radii, and the density ratios are uniformly bounded. Hence Proposition 2.1 applies with constants independent of $j$ . After enlarging $C_{0}$ and decreasing $r_{0}$ , estimate (3.2) holds uniformly for the approximating problems. The uniform $C^{1,\alpha_{0}}$ estimate then implies

(3.10)

Du_{j}\longrightarrow Du\qquad\text{locally uniformly in }\mathbb{R}^{n}.

For fixed $x_{0}\in\overline{\Omega}$ , set $y_{j}=Du_{j}(x_{0})$ and

\Psi_{j}(x)=(x-x_{0})\cdot(Du_{j}(x)-y_{j}).

Then $\Psi_{j}\to\Psi_{x_{0}}$ uniformly on $\overline{\Omega}$ .

Since $t\mapsto p_{\theta}(t)$ is nondecreasing, each positive level set of

\Psi_{x_{0}}(t\theta)=t\,p_{\theta}(t)

meets every admissible ray in at most one point. Hence every positive level set of $\Psi_{x_{0}}$ has Lebesgue measure zero by Fubini’s theorem in polar coordinates. Since the boundary of a convex domain has Lebesgue measure zero, dominated convergence implies, for every $h>0$ ,

(3.11)

\int_{\mathbb{R}^{n}}\mathbf{1}_{\Omega_{j}}\mathbf{1}_{\{\Psi_{j}<h\}}f_{j}\,\,\mathrm{d}x\longrightarrow\int_{\mathbb{R}^{n}}\mathbf{1}_{\Omega}\mathbf{1}_{\{\Psi_{x_{0}}<h\}}f\,\,\mathrm{d}x.

The smooth estimate (3.3) for $(\Omega_{j},\Omega_{j}^{*},f_{j},g_{j},u_{j})$ is uniform in $j$ . Passing to the limit in the integrated inequality proves (3.3) for arbitrary bounded convex domains.

∎

4. Blow-up analysis

We work directly with a blow-up sequence of the original transport problem. Fix $(x_{0},y_{0})\in\overline{\Omega}\times\overline{\Omega^{*}}$ with $y_{0}=Du(x_{0})$ , and let $h_{j}>0$ satisfy $h_{j}\to 0$ . For each $j$ , let $A_{j}$ be the positive definite linear map such that

B_{1}\subset A_{j}\bigl(S_{h_{j}}^{c}[u](x_{0})-x_{0}\bigr)\subset B_{n^{3/2}}.

Thus $A_{j}(S_{h_{j}}^{c}[u](x_{0})-x_{0})$ is in John position. Set

\tilde{x}=A_{j}(x-x_{0}),\qquad\tilde{y}=h_{j}^{-1}A_{j}^{-t}(y-y_{0}),

and

\Omega_{j}=A_{j}(\Omega-x_{0}),\qquad\Omega_{j}^{*}=h_{j}^{-1}A_{j}^{-t}(\Omega^{*}-y_{0}).

The normalized potentials are

	$\displaystyle\tilde{u}_{j}(\tilde{x})$	$\displaystyle=h_{j}^{-1}\bigl[\overline{u}(x_{0}+A_{j}^{-1}\tilde{x})-\overline{u}(x_{0})-y_{0}\cdot A_{j}^{-1}\tilde{x}\bigr],$
	$\displaystyle\tilde{v}_{j}(\tilde{y})$	$\displaystyle=h_{j}^{-1}\bigl[\overline{v}(y_{0}+h_{j}A_{j}^{t}\tilde{y})-\overline{v}(y_{0})-x_{0}\cdot h_{j}A_{j}^{t}\tilde{y}\bigr].$

At corresponding points,

\tilde{y}=D\tilde{u}_{j}(\tilde{x}),\qquad\tilde{x}=D\tilde{v}_{j}(\tilde{y}).

After multiplying the two measures by the same constant $a_{j}>0$ , set

\rho_{j}(\tilde{x})=a_{j}|\det A_{j}|^{-1}f(x_{0}+A_{j}^{-1}\tilde{x}),\qquad\rho_{j}^{*}(\tilde{y})=a_{j}h_{j}^{n}|\det A_{j}|g(y_{0}+h_{j}A_{j}^{t}\tilde{y}),

and

\,\mathrm{d}\mu_{j}=\rho_{j}\mathbf{1}_{\Omega_{j}}\,\,\mathrm{d}x,\qquad\,\mathrm{d}\nu_{j}=\rho_{j}^{*}\mathbf{1}_{\Omega_{j}^{*}}\,\,\mathrm{d}y.

Then the minimal convex extensions $\tilde{u}_{j},\tilde{v}_{j}$ satisfy

(4.1)

(D\tilde{u}_{j})_{\#}\mu_{j}=\nu_{j},\qquad\tilde{u}_{j}=(\tilde{v}_{j}+I_{\overline{\Omega_{j}^{*}}})^{*},\qquad\tilde{v}_{j}=(\tilde{u}_{j}+I_{\overline{\Omega_{j}}})^{*}.

Here $I_{E}$ denotes the convex indicator of $E$ . For a positive function $\rho$ on a set $E$ with $\inf_{E}\rho>0$ , define

\operatorname{osc}^{\rm rel}_{E}\rho=\frac{\sup_{E}\rho-\inf_{E}\rho}{\inf_{E}\rho}.

By (1.1),

(4.2)

\frac{\sup_{\Omega_{j}}\rho_{j}}{\inf_{\Omega_{j}}\rho_{j}}+\frac{\sup_{\Omega_{j}^{*}}\rho_{j}^{*}}{\inf_{\Omega_{j}^{*}}\rho_{j}^{*}}\leq D_{0}.

Here one may take $D_{0}=2\Lambda/\lambda$ . By [18, Lemma 2.6 and Corollary 2.5], the measures $\mu_{j}$ and $\nu_{j}$ have a common affine doubling constant depending only on $n,D_{0}$ . Moreover, by (2.4), we have $\tilde{u}_{j},\tilde{v}_{j}\in C^{1}_{\rm loc}(\mathbb{R}^{n}).$

Under the above normalization, we assume throughout that

(4.3)

\begin{gathered}0\in\overline{\Omega_{j}}\cap\overline{\Omega_{j}^{*}},\qquad\tilde{u}_{j}(0)=\tilde{v}_{j}(0)=0,\qquad D\tilde{u}_{j}(0)=D\tilde{v}_{j}(0)=0,\\ K_{j}:=S_{1}^{c}[\tilde{u}_{j}](0)=\{x:\tilde{u}_{j}(x)-p_{j}\cdot x<1\},\qquad B_{1}\subset K_{j}\subset B_{n^{3/2}}.\end{gathered}

The barycenter of $K_{j}$ is the origin. We also assume

(4.4)

\mu_{j}(D_{1}(\tilde{u}_{j},0))=\nu_{j}(D_{1}(\tilde{v}_{j},0))=1,

and, for every fixed $R<\infty$ ,

(4.5)

\operatorname{osc}^{\rm rel}_{D_{R}(\tilde{u}_{j},0)}\rho_{j}+\operatorname{osc}^{\rm rel}_{D_{R}(\tilde{v}_{j},0)}\rho_{j}^{*}\longrightarrow 0.

Finally, assume that

(4.6)

r^{-n}\mu_{j}(D_{r}(\tilde{u}_{j},0))\longrightarrow 1

uniformly for $r$ in compact subsets of $(0,\infty)$ . The arguments below use only (4.1)–(4.6), and therefore apply to any normalized sequence satisfying these conditions.

The inclusions (2.5) and (2.6), together with Lemmas 2.2 and 2.3, remain valid after normalization. We also use the following affine form of the subgradient image estimate. If $K=S_{h}^{c}[\tilde{u}_{j}](0)$ and $A_{h}K$ is in John position, then

(4.7)

B_{c}\subset h^{-1}A_{h}^{-t}\bigl(\partial\tilde{u}_{j}(K)-p_{h}\bigr)\subset B_{C},

where $c,C$ depend only on $n,D_{0}$ [6, Corollary 2.2].

The first consequence is uniform compactness at every fixed extrinsic scale.

Proposition 4.1.

For every fixed $0<R<\infty$ there is $C_{R}<\infty$ , independent of $j$ , such that whenever $x\in D_{R}(\tilde{u}_{j},0)$ and $y=D\tilde{u}_{j}(x)$ , one has

(4.8)

|x|+|y|\leq C_{R}.

Consequently, suppose that

(4.9)

\sup_{x\in D_{R}(\tilde{u}_{j},0)}\operatorname{dist}(x,L)\longrightarrow 0

for a linear subspace $L$ . Then

(4.10)

\sup_{\begin{subarray}{c}x\in D_{R}(\tilde{u}_{j},0)\\ y=D\tilde{u}_{j}(x)\end{subarray}}\left|P_{L^{\perp}}x\cdot P_{L^{\perp}}y\right|\longrightarrow 0.

Proof.

By (2.6) and (2.5)

D_{R}(\tilde{u}_{j},0)\subset S_{R^{2}}[\tilde{u}_{j}](0)\subset S^{c}_{CR^{2}}[\tilde{u}_{j}](0)\cap\Omega_{j}.

Lemma 2.3, used in both directions, shows that the last centered sub-level set has inner and outer radii bounded in terms of $R$ and the fixed data. Lemma 2.2 bounds $p_{h}$ . Applying (4.7) to that section bounds every corresponding subgradient $y$ . This proves (4.8). Condition (4.9) gives $P_{L^{\perp}}x\to 0$ uniformly on $D_{R}(\tilde{u}_{j},0)$ , and (4.10) follows from the bound for $y$ . ∎

We next choose bounded truncations that contain a prescribed extrinsic ball and remain uniformly separated from the truncating level.

Lemma 4.2.

Under (4.3), fix $R<\infty$ . There are $H=H(R)>R^{2}$ , $R_{H}<\infty$ , and $d_{H}>0$ , independent of $j$ , such that the convex sets

\Omega_{j,H}:=\Omega_{j}\cap\{\tilde{u}_{j}<H\}

are uniformly bounded, and satisfy

D_{R}(\tilde{u}_{j},0)\subset\Omega_{j,H}\subset D_{R_{H}}(\tilde{u}_{j},0),\qquad\operatorname{dist}\bigl(D_{R}(\tilde{u}_{j},0),\Omega_{j}\cap\{\tilde{u}_{j}=H\}\bigr)\geq d_{H}.

Proof.

After subtracting a supporting plane at the origin, $\tilde{u}_{j}\geq 0$ . For $y=D\tilde{u}_{j}(x)$ , convexity gives

(4.11)

\tilde{u}_{j}(x)\leq x\cdot y.

Consequently

D_{R}(\tilde{u}_{j},0)\subset\Omega_{j}\cap\{\tilde{u}_{j}<R^{2}\}.

Fix $H=R^{2}+1$ . By (2.6),

\Omega_{j}\cap\{\tilde{u}_{j}<H+1\}\subset S^{c}_{C(H+1)}[\tilde{u}_{j}](0)\cap\Omega_{j}.

Lemma 2.3 and (4.3) give uniform inner and outer radii for the centered sub-level set on the right. Lemma 2.2 and (4.7) then imply

(4.12)

\sup_{\begin{subarray}{c}x\in\Omega_{j}\\ \tilde{u}_{j}(x)\leq H\end{subarray}}\bigl(|x|+|D\tilde{u}_{j}(x)|\bigr)\leq L_{H}

with $L_{H}$ independent of $j$ . In particular, $\Omega_{j,H}$ is uniformly bounded.

If $x\in D_{R}(\tilde{u}_{j},0)$ and $z\in\Omega_{j}\cap\{\tilde{u}_{j}=H\}$ , then $[x,z]\subset\Omega_{j}\cap\{\tilde{u}_{j}\leq H\}$ . By (4.12), $\tilde{u}_{j}$ is $L_{H}$ -Lipschitz on this segment, while (4.11) implies

L_{H}|z-x|\geq\tilde{u}_{j}(z)-\tilde{u}_{j}(x)>H-R^{2}=1.

Thus $d_{H}=L_{H}^{-1}$ works. Finally, for $x\in\Omega_{j,H}$ , (4.12) implies

x\cdot D\tilde{u}_{j}(x)\leq L_{H}^{2}.

Taking $R_{H}^{2}>L_{H}^{2}$ proves $\Omega_{j,H}\subset D_{R_{H}}(\tilde{u}_{j},0)$ . ∎

The following elementary lemma identifies the support of a weak limit of measures supported in uniformly bounded convex sets.

Lemma 4.3.

Let $E_{j}\subset B_{R}\subset\mathbb{R}^{n}$ be nonempty open convex sets and $c_{j}>0$ . Suppose

(4.13)

\eta_{j}:=c_{j}\mathbf{1}_{E_{j}}\,\,\mathrm{d}x\rightharpoonup\eta,\qquad 0<\liminf_{j}\eta_{j}(E_{j})\leq\limsup_{j}\eta_{j}(E_{j})<\infty.

After passage to a subsequence, let $\overline{E}_{j}\to E$ in Hausdorff distance. Then

(4.14)

E=\operatorname{spt}\eta.

The following localized formulation will be used. Suppose, in addition, that $f_{j}$ is a continuous density on $E_{j}$ ,

\varepsilon_{j}:=\sup_{E_{j}}\left|\frac{f_{j}}{c_{j}}-1\right|\longrightarrow 0,\qquad\zeta_{j}:=f_{j}\mathbf{1}_{E_{j}}\,\,\mathrm{d}x\rightharpoonup\zeta.

If the mass bounds in (4.13) hold, then

\eta_{j}\rightharpoonup\zeta,\qquad\overline{E}_{j}\longrightarrow\operatorname{spt}\zeta\quad\text{in Hausdorff distance}.

Proof.

The inclusion $\operatorname{spt}\eta\subset E$ is immediate from Hausdorff convergence. Conversely, fix $x\in E$ and choose $x_{j}\in\overline{E}_{j}$ with $x_{j}\to x$ . Replacing $x_{j}$ by an interior point at distance $o(1)$ does not change the argument. Put $D=2R$ and fix $0<\rho<D$ . For $t=\rho/(4D)$ , by convexity, we have

x_{j}+t(E_{j}-x_{j})\subset E_{j}\cap B_{\rho/2}(x)

for all large $j$ . Hence

\eta_{j}(B_{\rho/2}(x))\geq t^{n}\eta_{j}(E_{j}).

Testing against a continuous cutoff which equals one on $B_{\rho/2}(x)$ and is supported in $B_{\rho}(x)$ , and using (4.13), gives $\eta(B_{\rho}(x))>0$ . Since $\rho$ is arbitrary, $x$ belongs to $\operatorname{spt}\eta$ , proving (4.14).

Moreover,

\|\zeta_{j}-\eta_{j}\|_{\rm TV}\leq\varepsilon_{j}\eta_{j}(E_{j})\longrightarrow 0,

so $\eta_{j}\rightharpoonup\zeta$ . Every Hausdorff-convergent subsequence of $\overline{E}_{j}$ has limit $\operatorname{spt}\zeta$ by the first part. By Blaschke compactness, we have

\overline{E}_{j}\longrightarrow\operatorname{spt}\zeta

for the whole sequence. ∎

5. Limits of normalized transports

We pass to a limit in the normalized problems of Section 4, identify the limiting marginals, and reduce each lower-dimensional source limit to an optimal transport problem on its affine hull. The exact mass identity is preserved under this reduction. For an extended-valued convex function, $\operatorname{cl}$ denotes its lower-semicontinuous closure. We first extract the limiting potentials and a locally finite transport plan.

Lemma 5.1.

Let a sequence of normalized transport problems satisfy (4.3)–(4.6). After passing to a subsequence there are a finite convex function $\tilde{u}_{\infty}:\mathbb{R}^{n}\to\mathbb{R}$ , its Legendre transform

\tilde{v}_{\infty}:=\tilde{u}_{\infty}^{*},

and a locally finite transport plan $\pi$ with the following properties.

(i)

(5.1)

\tilde{u}_{j}\longrightarrow\tilde{u}_{\infty}\qquad\text{locally uniformly in }\mathbb{R}^{n}.

Moreover, $\tilde{u}_{\infty},\tilde{v}_{\infty}\geq 0,\tilde{u}_{\infty}(0)=\tilde{v}_{\infty}(0)=0$ , and

(5.2)

\tilde{v}_{\infty}=\operatorname{cl}\bigl((\tilde{v}_{\infty}|_{\Omega_{\infty}^{*}})+I_{\Omega_{\infty}^{*}}\bigr),\qquad\tilde{u}_{\infty}=\tilde{v}_{\infty}^{*}.

Let $\Omega_{\infty}^{*}:=\operatorname{int}(\operatorname{dom}\tilde{v}_{\infty}).$ Then, $\Omega_{\infty}^{*}\neq\varnothing$ , and for every $Q\Subset\Omega_{\infty}^{*}$ , we have

(5.3)

Q\subset\Omega_{j}^{*}\quad\text{for all large }j,\qquad\tilde{u}_{j}^{*}\longrightarrow\tilde{v}_{\infty}\quad\text{uniformly on }Q.

(ii)

Consequently, for every $h>0$ ,

(5.4) $\overline{S_{h}^{c}[\tilde{u}_{j}](0)}\longrightarrow\overline{S_{h}^{c}[\tilde{u}_{\infty}](0)}$

in Hausdorff distance.

(iii)

Set $\pi_{j}=(\operatorname{id},D\tilde{u}_{j})_{\#}\mu_{j}$ and restrict each $\pi_{j}$ to $G_{R}:=\{(x,y)\in\mathbb{R}^{n}\times\mathbb{R}^{n}:x\cdot y<R^{2}\}.$ Then, for every $R>0$ ,

(5.5)

\pi_{j}\lfloor G_{R}\rightharpoonup\pi\lfloor G_{R}.

Moreover,

(5.6)

\operatorname{spt}\pi\subset\{(x,y):y\in\partial\tilde{u}_{\infty}(x)\}=\{(x,y):x\in\partial\tilde{v}_{\infty}(y)\},\qquad\pi(G_{R})=R^{n},\qquad\pi\{x\cdot y=R^{2}\}=0.

Proof.

For $h>0$ , set $K_{j,h}=S_{h}^{c}[\tilde{u}_{j}](0)$ , and let $p_{j,h}$ be the slope of its centering plane. Thus $K_{j,1}=K_{j}$ and $p_{j,1}=p_{j}$ in the notation of (4.3). By (4.1), the Alexandrov measure of $\tilde{u}_{j}$ satisfies

\,\mathrm{d}M_{\tilde{u}_{j}}=\frac{\rho_{j}}{\rho_{j}^{*}(D\tilde{u}_{j})}\mathbf{1}_{\Omega_{j}}\,\,\mathrm{d}x.

Thus $\operatorname{spt}M_{\tilde{u}_{j}}=\overline{\Omega_{j}}$ , and (4.2) gives a common affine doubling bound. Since $0\in\overline{\Omega_{j}}$ by (4.3), the geometric decay of centered sections [6, Lemma 2.2] yields $s\in(0,1)$ , independent of $j$ , such that

K_{j,sh}\subset\tfrac{1}{2}K_{j,h}\qquad(h>0).

Consequently, for every $R>1$ , there is $H_{R}<\infty$ , independent of $j$ , such that

(5.7)

RK_{j}\subset K_{j,H_{R}}.

Since $B_{1}\subset K_{j}$ by (4.3), (5.7) gives $B_{R}\subset K_{j,H_{R}}$ . Convexity and (4.3) imply $\tilde{u}_{j}\geq 0$ , while Lemma 2.2 yields

|p_{j,H_{R}}|\leq\frac{H_{R}}{R}.

Hence

(5.8)

0\leq\tilde{u}_{j}(x)\leq H_{R}+p_{j,H_{R}}\cdot x\leq 2H_{R}\qquad(x\in B_{R}).

Convexity and (5.8), applied on a slightly larger ball, yield uniform local Lipschitz bounds. Arzelà–Ascoli and a diagonal argument prove (5.1). Since $\tilde{u}_{j}\geq 0$ and $\tilde{u}_{j}(0)=0$ , we have $\tilde{u}_{\infty}\geq 0$ and $\tilde{u}_{\infty}(0)=0$ . Therefore

\tilde{v}_{\infty}(0)=-\inf_{\mathbb{R}^{n}}\tilde{u}_{\infty}=0,\qquad\tilde{v}_{\infty}\geq 0.

The first identity in (5.2) follows from [23, Theorem 7.5]. The second follows from the definition of $\tilde{v}_{\infty}$ and the biconjugation theorem [23, Theorem 12.2].

To see that $\Omega_{\infty}^{*}\neq\varnothing$ , after passing to a further subsequence, assume that $p_{j}\to p$ . Taking $h=1$ in (4.7), we obtain

p_{j}+B_{c}\subset\partial\tilde{u}_{j}(K_{j}).

Fix $y\in p+B_{c/2}$ . For all large $j$ , there is $x_{j}\in K_{j}$ such that $y\in\partial\tilde{u}_{j}(x_{j})$ . Since $K_{j}\subset B_{n^{3/2}}$ and $\tilde{u}_{j}\geq 0$ ,

\tilde{u}_{j}^{*}(y)=x_{j}\cdot y-\tilde{u}_{j}(x_{j})\leq C.

Hence, for every $z\in\mathbb{R}^{n}$ ,

z\cdot y-\tilde{u}_{j}(z)\leq C.

Letting $j\to\infty$ and then taking the supremum over $z$ , we obtain

\tilde{v}_{\infty}(y)=\tilde{u}_{\infty}^{*}(y)\leq C.

Thus

p+B_{c/2}\subset\operatorname{dom}\tilde{v}_{\infty},

and therefore $\Omega_{\infty}^{*}\neq\varnothing$ .

Fix $Q\Subset\Omega_{\infty}^{*}$ , and choose $\delta>0$ such that

\overline{Q}+2\delta\overline{B}_{1}\Subset\Omega_{\infty}^{*}.

Since $\tilde{v}_{\infty}$ is continuous in $\Omega_{\infty}^{*}$ ,

M:=\sup_{\overline{Q}+2\delta\overline{B}_{1}}\tilde{v}_{\infty}<\infty.

For $y\in\overline{Q}+\delta\overline{B}_{1}$ , $x\neq 0$ , and $z=y+\delta x/|x|$ , Fenchel’s inequality implies

(5.9)

\tilde{u}_{\infty}(x)-y\cdot x\geq\delta|x|-M.

Choose $R$ so large that $\delta R-M\geq 2$ . By (5.1) and (5.9), for all large $j$ ,

\tilde{u}_{j}(Re)-y\cdot Re\geq 1

for every $e\in\mathbb{S}^{n-1}$ and $y\in\overline{Q}+\delta\overline{B}_{1}$ . Convexity and $\tilde{u}_{j}(0)=0$ then imply

\tilde{u}_{j}(te)-y\cdot te\geq\frac{t}{R}\qquad(t\geq R).

Thus the suprema defining $\tilde{u}_{j}^{*}$ and $\tilde{v}_{\infty}$ on $\overline{Q}+\delta\overline{B}_{1}$ are attained in one fixed ball. It follows that

\sup_{\overline{Q}+\delta\overline{B}_{1}}|\tilde{u}_{j}^{*}-\tilde{v}_{\infty}|\leq\sup_{\overline{B}_{R}}|\tilde{u}_{j}-\tilde{u}_{\infty}|\longrightarrow 0.

In particular,

\overline{Q}+\delta\overline{B}_{1}\subset\operatorname{dom}\tilde{u}_{j}^{*}=\overline{\Omega_{j}^{*}}.

Since $\Omega_{j}^{*}$ is open and convex, $Q\subset\Omega_{j}^{*}$ , which proves (5.3).

For each fixed $h>0$ , (4.3) and Lemma 2.3 give

(5.10)

B_{c_{h}}\subset K_{j,h}\subset B_{C_{h}},

with constants independent of $j$ . Hence Lemma 2.2 bounds the centering slopes, and (5.4) follows directly from (5.1), preservation of the barycenter under Hausdorff convergence, and uniqueness of the centering plane.

We now prove (5.5) and (5.6). For every $R>0$ , by (4.6), we have

\pi_{j}(G_{R})=\mu_{j}\bigl(D_{R}(\tilde{u}_{j},0)\bigr)\longrightarrow R^{n}.

Moreover, by Proposition 4.1, there is a compact set $C_{R}\subset\mathbb{R}^{n}\times\mathbb{R}^{n}$ , independent of $j$ , such that

\operatorname{spt}(\pi_{j}\lfloor G_{R})\subset C_{R}.

Thus the measures $\pi_{j}$ have uniformly bounded mass on every compact subset of $\mathbb{R}^{n}\times\mathbb{R}^{n}$ . After passing to a diagonal subsequence, there is a locally finite measure $\pi$ such that

\int\varphi\,\,\mathrm{d}\pi_{j}\longrightarrow\int\varphi\,\,\mathrm{d}\pi\qquad\text{for every }\varphi\in C_{c}(\mathbb{R}^{n}\times\mathbb{R}^{n}).

We first prove the support assertion in (5.6). Suppose that

(x_{j},y_{j})\longrightarrow(x,y),\qquad y_{j}\in\partial\tilde{u}_{j}(x_{j}).

For every $z\in\mathbb{R}^{n}$ ,

\tilde{u}_{j}(z)\geq\tilde{u}_{j}(x_{j})+y_{j}\cdot(z-x_{j}).

Passing to the limit using (5.1), we obtain

\tilde{u}_{\infty}(z)\geq\tilde{u}_{\infty}(x)+y\cdot(z-x),

and hence $y\in\partial\tilde{u}_{\infty}(x)$ . Therefore

\operatorname{spt}\pi\subset\{(x,y):y\in\partial\tilde{u}_{\infty}(x)\}.

Since $\tilde{v}_{\infty}=\tilde{u}_{\infty}^{*}$ , by Legendre duality, we have

y\in\partial\tilde{u}_{\infty}(x)\quad\Longleftrightarrow\quad x\in\partial\tilde{v}_{\infty}(y).

This proves the first assertion in (5.6).

It remains to identify the mass at every radius. At finite $j$ , each positive level of $(x,y)\mapsto x\cdot y$ has zero $\pi_{j}$ -mass by the radial argument used in the proof of (3.11). Let $0<R<k$ and $0<\varepsilon<\min\{R,k-R\}$ . Weak convergence of the restrictions at radius $k$ yields that

	$\displaystyle\pi\{x\cdot y<R^{2}\}$	$\displaystyle\geq(R-\varepsilon)^{n},$
	$\displaystyle\pi\{x\cdot y\leq R^{2}\}$	$\displaystyle\leq(R+\varepsilon)^{n}.$

Letting $\varepsilon\rightarrow 0$ proves

(5.11)

\pi(G_{R})=R^{n},\qquad\pi\{x\cdot y=R^{2}\}=0.

The restrictions obtained from different integer radii agree on their common continuity sets. They therefore define one locally finite plan $\pi$ , and (5.5)–(5.6) follow. ∎

The next lemma gives a direct geometric description of the density created by a lower-dimensional collapse. In the following, $\operatorname{aff}(E)$ denotes the affine hull of $E$ , i.e., the smallest affine subspace containing $E$ .

Lemma 5.2.

Let $E_{j}\subset B_{R}\subset\mathbb{R}^{n}$ be nonempty open convex sets and let $c_{j}>0$ . Suppose that

\eta_{j}:=c_{j}\mathbf{1}_{E_{j}}\,\,\mathrm{d}x\rightharpoonup\eta,\qquad 0<\liminf_{j}\eta_{j}(E_{j})\leq\limsup_{j}\eta_{j}(E_{j})<\infty,

and that

\overline{E}_{j}\longrightarrow E=\operatorname{spt}\eta\qquad\text{in Hausdorff distance}.

Set

L=\operatorname{aff}E,\qquad m=\dim L,\qquad\Omega^{\prime}=\operatorname{relint}E.

If $m=n$ , then

\eta=c\mathbf{1}_{\Omega^{\prime}}\,\,\mathrm{d}x

for some constant $c>0$ . If $0<m<n$ and $q=n-m$ , then

(5.12)

\eta=\rho\mathbf{1}_{\Omega^{\prime}}\,\,\mathrm{d}\mathcal{H}^{m}\lfloor L,\qquad\rho^{1/q}\ \text{is positive and concave on }\Omega^{\prime}.

If $m=0$ , then $\eta$ is a positive multiple of a Dirac mass.

Proof.

The case $m=0$ is immediate. After a translation and a rotation, assume for the remaining cases that

L=\mathbb{R}^{m}\times\{0\},

and denote the orthogonal projection onto $L$ by $P_{L}$ .

Suppose first that $m=n$ . Hausdorff convergence of bounded convex sets gives

|E_{j}\triangle E|\longrightarrow 0,\qquad|E_{j}|\longrightarrow|E|>0.

Since all the measures are supported in $B_{R}$ , weak convergence also gives

c_{j}|E_{j}|=\eta_{j}(\mathbb{R}^{n})\longrightarrow\eta(\mathbb{R}^{n}).

Thus $c_{j}\to c>0$ , and hence $\eta=c\mathbf{1}_{\Omega^{\prime}}\,\,\mathrm{d}x,$ because the boundary of a full-dimensional convex set has Lebesgue measure zero.

Now let $0<m<n$ and put $q=n-m$ . Set

F_{j,x}:=\{z\in L^{\perp}:x+z\in E_{j}\},\qquad x\in P_{L}E_{j}.

By Fubini,

(5.13)

(P_{L})_{\#}\eta_{j}=g_{j}\,\,\mathrm{d}\mathcal{H}^{m}\lfloor L,\qquad g_{j}(x):=c_{j}\mathcal{H}^{q}(F_{j,x}).

For $x_{0},x_{1}\in P_{L}E_{j}$ and $0<t<1$ , convexity gives

(1-t)F_{j,x_{0}}+tF_{j,x_{1}}\subset F_{j,(1-t)x_{0}+tx_{1}}.

The $q$ -dimensional Brunn–Minkowski inequality therefore shows that

(5.14)

g_{j}^{1/q}\quad\text{is nonnegative and concave on }P_{L}E_{j}.

Since $E\subset L$ and $\overline{E}_{j}\to E$ ,

\delta_{j}:=\sup_{z\in E_{j}}|z-P_{L}z|\longrightarrow 0.

Consequently, for every $\varphi\in C_{c}(\mathbb{R}^{n})$ ,

	$\displaystyle\left\|\int\varphi\,\,\mathrm{d}\eta_{j}-\int_{L}\varphi(x)g_{j}(x)\,\,\mathrm{d}\mathcal{H}^{m}(x)\right\|$	$\displaystyle\leq\eta_{j}(E_{j})\sup_{\|z-z^{\prime}\|\leq\delta_{j}}\|\varphi(z)-\varphi(z^{\prime})\|$
		$\displaystyle\longrightarrow 0.$

Thus the projected measures in (5.13) converge weakly to $\eta$ .

We next exclude concentration on an $\mathcal{H}^{m}$ -null subset of $L$ . Since $P_{L}E_{j}\to E$ in Hausdorff distance, there are $x_{0}\in\Omega^{\prime}$ and $r>0$ such that $B_{r}^{L}(x_{0})\subset P_{L}E_{j}$ for all sufficiently large $j$ . Fix $x\in P_{L}E_{j}$ . For every $y\in B_{r}^{L}(x_{0})$ , concavity and nonnegativity give

g_{j}^{1/q}\!\left(\frac{x+y}{2}\right)\geq\frac{g_{j}^{1/q}(x)+g_{j}^{1/q}(y)}{2}\geq\frac{g_{j}^{1/q}(x)}{2},

where $B_{r}^{L}(x_{0})$ denotes a ball in $L$ with radius $r$ and center $x_{0}.$ By convexity

B_{\frac{r}{2}}^{L}(\frac{x+x_{0}}{2})=\{\frac{x+y}{2}:y\in B_{r}^{L}(x_{0})\}\subset P_{L}E_{j}.

Hence

\eta_{j}(E_{j})=\int_{P_{L}E_{j}}g_{j}\,\,\mathrm{d}\mathcal{H}^{m}\geq\omega_{m}\left(\frac{r}{2}\right)^{m}\frac{g_{j}(x)}{2^{q}}.

The uniform upper mass bound therefore yields

(5.15)

\sup_{P_{L}E_{j}}g_{j}\leq C.

In particular, $(P_{L})_{\#}\eta_{j}\leq C\mathcal{H}^{m}\lfloor L.$ Passing to the weak limit gives $\eta\leq C\mathcal{H}^{m}\lfloor L.$ Thus no mass can concentrate on a set of zero $\mathcal{H}^{m}$ -measure.

By (5.15), and concavity of $g_{j}^{1/q}$ , after passing to a subsequence,

g_{j}^{1/q}\longrightarrow w\qquad\text{locally uniformly on }\Omega^{\prime},

for some finite, nonnegative, concave function $w.$ By (5.13) and the weak convergence just proved,

\eta\lfloor\Omega^{\prime}=w^{q}\,\,\mathrm{d}\mathcal{H}^{m}\lfloor\Omega^{\prime}.

Since $\eta\leq C\mathcal{H}^{m}\lfloor L$ , $\operatorname{spt}\eta=E$ , and the relative boundary of a convex set has zero $\mathcal{H}^{m}$ -measure, it follows that

\eta=w^{q}\mathbf{1}_{\Omega^{\prime}}\,\,\mathrm{d}\mathcal{H}^{m}\lfloor L.

The function $w$ is not identically zero. By concavity and nonnegativity we have $w>0$ on $\Omega^{\prime}$ . Taking $\rho=w^{q}$ proves (5.12). ∎

The next lemma identifies the source marginal and proves local convergence of the source supports to its affine hull. In particular, the zero-dimensional alternative does not occur.

Lemma 5.3.

Let $\mu$ be the first marginal of the limiting transport plan $\pi$ , which is obtained in Lemma 5.1. Then $\mu$ is nonzero and locally finite, $0\in\operatorname{spt}\mu$ , and $\operatorname{spt}\mu$ is convex. If

L=\operatorname{aff}(\operatorname{spt}\mu),\qquad m=\dim L,

then $L$ is a linear subspace and $1\leq m\leq n$ . Moreover, for every fixed $R<\infty$ ,

(5.16)

\sup_{x\in D_{R}(\tilde{u}_{j},0)}\operatorname{dist}(x,L)\longrightarrow 0.

The source marginal has one of the following forms.

(a)

If $m=n$ , then

$\mu=\rho_{\infty}\mathbf{1}_{\Omega_{\infty}}\,\,\mathrm{d}x$

for a constant $\rho_{\infty}>0$ and the open convex set $\Omega_{\infty}=\operatorname{int}(\operatorname{spt}\mu)$ .

(b)

If $1\leq m<n$ , then, with $q=n-m$ ,

(5.17)

\mu=\rho(x)\mathbf{1}_{\Omega_{\infty}}\,\,\mathrm{d}\mathcal{H}^{m}\lfloor L,\qquad\rho^{1/q}\ \text{is positive and concave on }\Omega_{\infty},

where $\Omega_{\infty}=\operatorname{relint}(\operatorname{spt}\mu)$ .

Proof.

We first prove local finiteness. By (4.3), Lemma 2.3, and (2.6), there is a dimensional constant $r_{*}>0$ such that

\Omega_{j}\cap B_{r_{*}}\subset S_{1}[\tilde{u}_{j}](0).

Together with (2.5), this yields

\tfrac{1}{2}(\Omega_{j}\cap B_{r_{*}})\subset D_{1}(\tilde{u}_{j},0).

Convexity, $0\in\overline{\Omega_{j}}$ , and (4.2) imply

\mu_{j}(B_{r_{*}})\leq 2^{n}D_{0}\,\mu_{j}\bigl(\tfrac{1}{2}(\Omega_{j}\cap B_{r_{*}})\bigr)\leq 2^{n}D_{0}.

The common doubling bound then shows that

\sup_{j}\mu_{j}(B_{M})<\infty\qquad(M<\infty).

Combining this with (5.5), we conclude that $\mu$ is locally finite.

Choose $1\leq R_{1}<R_{2}<\cdots\to\infty$ . By Lemma 4.2, the heights may be chosen increasingly so that

(5.18)

D_{R_{k}}(\tilde{u}_{j},0)\subset\Omega_{j,H_{k}}\subset D_{\widehat{R}_{k}}(\tilde{u}_{j},0),

where $\widehat{R}_{k}<\infty$ is independent of $j$ , and the sets $\Omega_{j,H_{k}}$ are nested in $k$ . Set

(5.19)

c_{k,j}:=\inf_{\Omega_{j,H_{k}}}\rho_{j}>0,\qquad\varepsilon_{k,j}:=\sup_{\Omega_{j,H_{k}}}\left(\frac{\rho_{j}}{c_{k,j}}-1\right)=\operatorname{osc}^{\rm rel}_{\Omega_{j,H_{k}}}\rho_{j}\longrightarrow 0\qquad(j\to\infty).

Here the convergence follows from $\Omega_{j,H_{k}}\subset D_{\widehat{R}_{k}}(\tilde{u}_{j},0)$ and (4.5). Define

(5.20)

\eta_{k,j}=c_{k,j}\mathbf{1}_{\Omega_{j,H_{k}}}\,\,\mathrm{d}x,\qquad\zeta_{k,j}=\mu_{j}\lfloor\Omega_{j,H_{k}}.

The two inclusions in (5.18), together with (4.4) and (4.6), give uniform positive lower and finite upper bounds for $\zeta_{k,j}(\mathbb{R}^{n})$ . Since

\eta_{k,j}\leq\zeta_{k,j}\leq(1+\varepsilon_{k,j})\eta_{k,j},

the same mass bounds hold for $\eta_{k,j}$ , and

\|\eta_{k,j}-\zeta_{k,j}\|_{\mathrm{TV}}\leq\varepsilon_{k,j}\eta_{k,j}(\mathbb{R}^{n})\longrightarrow 0.

After a diagonal extraction,

\eta_{k,j}\rightharpoonup\zeta_{k},\qquad\zeta_{k,j}\rightharpoonup\zeta_{k}

for every $k$ . Since the truncations are nested, we have

(5.21)

\zeta_{k}\leq\zeta_{\ell}\qquad(k\leq\ell).

By Lemma 4.3,

(5.22)

\overline{\Omega_{j,H_{k}}}\longrightarrow S_{k}:=\operatorname{spt}\zeta_{k}\qquad\text{in Hausdorff distance}

for every $k$ . Thus each $S_{k}$ is convex. Moreover, $0\in S_{k}$ , because $0\in\overline{\Omega_{j,H_{k}}}$ , and (5.21) gives $S_{k}\subset S_{\ell}$ for $k\leq\ell$ .

Let $\mu_{R}$ be the first marginal of $\pi\lfloor G_{R}$ . Taking weak limits in the measure inclusions induced by (5.18), and using Lemma 5.1 for the two outer terms, we obtain

(5.23)

\mu_{R_{k}}\leq\zeta_{k}\leq\mu_{\widehat{R}_{k}}.

Since $\mu_{R}\uparrow\mu$ as $R\to\infty$ , it follows that

(5.24)

\mu=\sup_{k}\zeta_{k}.

Consequently,

\operatorname{spt}\mu=\overline{\bigcup_{k}S_{k}}.

Hence $\operatorname{spt}\mu$ is convex, $0\in\operatorname{spt}\mu$ , and $L=\operatorname{aff}(\operatorname{spt}\mu)$ is a linear subspace.

For each $k$ , (5.18) and (5.22) give

\sup_{x\in D_{R_{k}}(\tilde{u}_{j},0)}\operatorname{dist}(x,L)\leq d_{H}(\overline{\Omega_{j,H_{k}}},S_{k})\longrightarrow 0.

Since $R_{k}\to\infty$ and the sets $D_{R}(\tilde{u}_{j},0)$ are nested in $R$ , this proves (5.16) for every fixed $R<\infty$ .

It remains to rule out $m=0$ . Suppose that $L=\{0\}$ . Since

\operatorname{spt}(\pi_{j}\lfloor G_{1})\subset\operatorname{spt}\pi_{j}\cap\{(x,y):x\cdot y\leq 1\},

(5.16) with $R=2$ gives

\sup_{(x,y)\in\operatorname{spt}(\pi_{j}\lfloor G_{1})}|x|\longrightarrow 0.

On the other hand, Proposition 4.1 gives

\sup_{j}\sup_{(x,y)\in\operatorname{spt}(\pi_{j}\lfloor G_{1})}|y|<\infty.

Consequently,

\sup_{(x,y)\in\operatorname{spt}(\pi_{j}\lfloor G_{1})}|x\cdot y|\longrightarrow 0.

Thus, for every fixed $0<r<1$ ,

\operatorname{spt}(\pi_{j}\lfloor G_{1})\subset G_{r}

for all sufficiently large $j$ . Since $\pi_{j}(G_{1})=1$ , this implies

\mu_{j}(D_{r}(\tilde{u}_{j},0))=\pi_{j}(G_{r})=1,

contradicting (4.6), because

\mu_{j}(D_{r}(\tilde{u}_{j},0))\longrightarrow r^{n}<1.

Therefore $m\geq 1$ .

Set $S=\operatorname{spt}\mu$ . Since $S=\overline{\bigcup_{k}S_{k}}$ and the sets $S_{k}$ are increasing, there is $k_{0}$ such that $\operatorname{aff}S_{k}=L$ for $k\geq k_{0}$ . Moreover, by convexity we have that every compact subset of $\operatorname{relint}S$ is contained in $\operatorname{relint}S_{k}$ for all sufficiently large $k$ . For $k\geq k_{0}$ , apply Lemma 5.2 to $E_{j}=\Omega_{j,H_{k}}$ and $\eta_{j}=\eta_{k,j}$ . Denote $\Omega_{k}=\operatorname{relint}S_{k}.$

If $m=n$ , set $\Omega_{\infty}=\operatorname{int}S$ . The lemma gives

\zeta_{k}=a_{k}\mathbf{1}_{\Omega_{k}}\,\,\mathrm{d}x,\qquad a_{k}>0.

By (5.21) we have that $a_{k}$ is nondecreasing. Choosing a ball $B\Subset\Omega_{k_{0}}$ , we have

a_{k}|B|=\zeta_{k}(B)\leq\mu(B),\qquad k\geq k_{0},

so $a_{k}\uparrow\rho_{\infty}<\infty$ . Therefore (5.24) gives

\mu=\rho_{\infty}\mathbf{1}_{\Omega_{\infty}}\,\,\mathrm{d}x.

Now assume $1\leq m<n$ and put $q=n-m$ . Lemma 5.2 gives

\zeta_{k}=\rho_{k}\mathbf{1}_{\Omega_{k}}\,\,\mathrm{d}\mathcal{H}^{m}\lfloor L,

with $\rho_{k}^{1/q}$ positive and concave on $\Omega_{k}$ . Since $S_{k}\subset S_{\ell}$ and $\zeta_{k}\leq\zeta_{\ell}$ , the continuous representatives satisfy

\rho_{k}^{1/q}\leq\rho_{\ell}^{1/q}\qquad\text{on }\Omega_{k},\qquad k\leq\ell.

Set $\Omega_{\infty}=\operatorname{relint}S$ . Then

w(x):=\lim_{k\to\infty}\rho_{k}^{1/q}(x),\qquad x\in\Omega_{\infty},

where the limit is taken for sufficiently large $k$ such that $x\in\Omega_{k}$ . It is well defined and concave, initially with values in $(0,\infty]$ . It is finite everywhere. Indeed, if $w(x_{0})=\infty$ , choose a ball $B\Subset\Omega_{\infty}$ and $0<t<1$ . For all sufficiently large $k$ , concavity gives

\rho_{k}^{1/q}((1-t)x_{0}+ty)\geq(1-t)\rho_{k}^{1/q}(x_{0})\qquad(y\in B).

Letting $k\to\infty$ would make $w$ infinite on the open set $(1-t)x_{0}+tB$ . Since $\mu=\sup_{k}\zeta_{k}$ , monotone convergence would then give infinite $\mu$ -mass on a compact subset of this set, contradicting local finiteness. Thus $w$ is positive and finite on $\Omega_{\infty}$ .

Extend $\rho_{k}$ by zero outside $\Omega_{k}$ . These densities increase almost everywhere to $w^{q}\mathbf{1}_{\Omega_{\infty}}$ . Hence monotone convergence in (5.24) yields that

\mu=w^{q}\mathbf{1}_{\Omega_{\infty}}\,\,\mathrm{d}\mathcal{H}^{m}\lfloor L.

Taking $\rho=w^{q}$ proves (5.17). ∎

We next identify the target marginal.

Lemma 5.4.

Assume (4.3)–(4.6), and suppose that, for an $m$ -dimensional subspace $L$ , with $1\leq m<n$ , (4.9) holds on $D_{R}(\tilde{u}_{j},0)$ for every fixed $R$ . Let $\mu$ and $\nu$ be the first and second marginals of $\pi$ , respectively, and write, as in (5.17),

\mu=\rho\,\mathbf{1}_{\Omega_{\infty}}\,\mathcal{H}^{m}\lfloor L,\qquad\Omega_{\infty}=\operatorname{relint}(\operatorname{spt}\mu).

After passing to a subsequence, the following statements hold.

(1)

Let $\Omega_{\infty}^{*}$ be as in Lemma 5.1. Then there is a constant $\rho_{\infty}^{*}>0$ such that

(5.25) $\nu=\rho_{\infty}^{*}\,\mathcal{L}^{n}\lfloor\Omega_{\infty}^{*}.$

Moreover, the normalized target measures converge in total variation on every compact subset of $\Omega_{\infty}^{*}$ to the measure in (5.25).

(2)

Set $\Omega_{L}^{*}=P_{L}\Omega_{\infty}^{*}.$ After identifying $L$ isometrically with $\mathbb{R}^{m}$ , there are dual convex potentials $\phi$ on $\Omega_{\infty}$ and $\varphi$ on $\Omega_{L}^{*}$ such that

(5.26)

\tilde{u}_{\infty}(x)=\phi(x),\qquad\tilde{v}_{\infty}(y)=\varphi(P_{L}y),\qquad x\in\Omega_{\infty},\ y\in\Omega_{\infty}^{*}.

(3)

The projected target density

\rho_{L}^{*}(\xi)=\rho_{\infty}^{*}\mathcal{H}^{q}\bigl(\Omega_{\infty}^{*}\cap(\xi+L^{\perp})\bigr)

is finite and positive for $\xi\in\Omega_{L}^{*},$ and $(\rho_{L}^{*})^{1/q}$ is concave in $\Omega_{L}^{*}$ . The minimal extensions satisfy

(D\phi)_{\#}\bigl(\rho\,\mathcal{H}^{m}\lfloor\Omega_{\infty}\bigr)=\rho_{L}^{*}\mathbf{1}_{\Omega_{L}^{*}}\,\,\mathrm{d}\xi,\qquad(D\varphi)_{\#}\bigl(\rho_{L}^{*}\mathbf{1}_{\Omega_{L}^{*}}\,\,\mathrm{d}\xi\bigr)=\rho\,\mathcal{H}^{m}\lfloor\Omega_{\infty}.

Moreover, for every $r>0$ ,

(5.27)

\int_{\{x\in\Omega_{\infty}:x\cdot D\phi(x)<r^{2}\}}\rho(x)\,\,\mathrm{d}\mathcal{H}^{m}(x)=\int_{\{\xi\in\Omega_{L}^{*}:\xi\cdot D\varphi(\xi)<r^{2}\}}\rho_{L}^{*}(\xi)\,\,\mathrm{d}\xi=r^{n}.

Proof.

Set

(5.28)

b_{j}:=\inf_{D_{1}(\tilde{v}_{j},0)}\rho_{j}^{*}.

We first prove uniform bounds for $b_{j}$ . With $K_{j}$ and $p_{j}$ as in (4.3), (2.7) and (4.3) give

(5.29)

p_{j}+B_{c_{0}}\subset\partial\tilde{u}_{j}(K_{j})\subset\overline{\Omega_{j}^{*}},\qquad|p_{j}|\leq 1,

for some constant $c_{0}$ depending only on $n.$ Hence,

(5.30)

p_{j}+B_{c_{0}/2}\subset\Omega_{j}^{*}.

For every $y$ in this ball, the corresponding point $x=D\tilde{v}_{j}(y)$ belongs to $K_{j}$ . The bounds on $p_{j}$ , $K_{j}$ , and the ball give $|x|+|y|\leq C$ , and therefore $x\cdot y<R_{0}^{2}$ for a fixed $R_{0}\geq 1$ . Thus

(5.31)

p_{j}+B_{c_{0}/2}\subset D_{R_{0}}(\tilde{v}_{j},0).

Consequently, by (4.5) we have

(5.32)

\sup_{D_{R_{0}}(\tilde{v}_{j},0)}\rho_{j}^{*}=(1+o(1))b_{j}.

The ball in (5.31) and $\nu_{j}(D_{R_{0}}(\tilde{v}_{j},0))=\mu_{j}(D_{R_{0}}(\tilde{u}_{j},0))\to R_{0}^{n}$ (follows by (4.6)) give $b_{j}\leq C$ . Conversely, Proposition 4.1 implies that $D_{1}(\tilde{v}_{j},0)$ is contained in a fixed Euclidean ball independent of $j$ , and

1=\nu_{j}(D_{1}(\tilde{v}_{j},0))\leq(1+o(1))b_{j}|D_{1}(\tilde{v}_{j},0)|\leq Cb_{j},

where the first equality follows from the assumption (4.4). This proves $0<c\leq b_{j}\leq C$ . Pass to a subsequence with $b_{j}\to\rho^{*}_{\infty}>0,$ for some constant $\rho^{*}_{\infty}.$

We next identify the density on compact subsets. Fix $Q\Subset\Omega_{\infty}^{*}$ , and choose $Q^{\prime}$ with $Q\Subset Q^{\prime}\Subset\Omega_{\infty}^{*}$ . By (5.3), for all large $j$ ,

(5.33)

Q^{\prime}\subset\Omega_{j}^{*},\qquad\tilde{v}_{j}\longrightarrow\tilde{v}_{\infty}\quad\hbox{uniformly on }Q^{\prime}.

By convexity we have

(5.34)

\sup_{y\in Q}|\partial\tilde{v}_{j}(y)|\leq C_{Q}.

Thus $|y\cdot D\tilde{v}_{j}(y)|\leq C_{Q}$ for every $y\in Q$ , so $Q\subset D_{R}(\tilde{v}_{j},0)$ for some fixed $R$ . By (4.5), we have

(5.35)

\sup_{Q}\left|\frac{\rho^{*}_{j}}{b_{j}}-1\right|\longrightarrow 0.

Hence the target marginals converge on every compact subset of $\Omega_{\infty}^{*}$ , indeed in local total variation, to $\rho^{*}_{\infty}\mathbf{1}_{\Omega_{\infty}^{*}}\,\,\mathrm{d}y$ .

It remains to identify the second marginal on the boundary of $\Omega_{\infty}^{*}$ . For $R>0$ , set

\nu_{j,R}:=(\operatorname{pr}_{2})_{\#}(\pi_{j}\lfloor G_{R}),\qquad\nu_{R}:=(\operatorname{pr}_{2})_{\#}(\pi\lfloor G_{R}).

By (5.5),

\nu_{j,R}\rightharpoonup\nu_{R}.

Moreover,

\nu_{j,R}=\rho_{j}^{*}\mathbf{1}_{D_{R}(\tilde{v}_{j},0)}\,\,\mathrm{d}y\leq C_{R}\,\,\mathrm{d}y,

where $C_{R}$ is independent of $j$ . Passing to the limit, we have

\nu_{R}\leq C_{R}\,\,\mathrm{d}y.

Since $G_{R}\uparrow\mathbb{R}^{n}\times\mathbb{R}^{n}$ , we have $\nu_{R}\uparrow\nu$ , where $\nu$ is the second marginal of $\pi$ . Consequently,

\nu\ll\mathcal{L}^{n}.

On the other hand, the support relation in (5.6) gives

\nu\bigl(\mathbb{R}^{n}\setminus\operatorname{dom}\tilde{v}_{\infty}\bigr)=0.

Since

\operatorname{dom}\tilde{v}_{\infty}\setminus\Omega_{\infty}^{*}\subset\partial\Omega_{\infty}^{*}

and the boundary of a full-dimensional convex set has Lebesgue measure zero, it follows that

\nu(\mathbb{R}^{n}\setminus\Omega_{\infty}^{*})=0.

Finally, let $\varphi\in C_{c}(\Omega_{\infty}^{*})$ . The local boundedness of the subgradients of $\tilde{v}_{j}$ and $\tilde{v}_{\infty}$ allows us to choose $R$ such that all the transport pairs whose second coordinate lies in $\operatorname{spt}\varphi$ belong to $G_{R}$ . Therefore, by (5.35),

\int\varphi\,\,\mathrm{d}\nu=\lim_{j\to\infty}\int\varphi(y)\rho_{j}^{*}(y)\,\,\mathrm{d}y=\rho_{\infty}^{*}\int\varphi(y)\,\,\mathrm{d}y.

Thus

\nu=\rho_{\infty}^{*}\mathbf{1}_{\Omega_{\infty}^{*}}\,\,\mathrm{d}y.

We now factorize the dual potential along the directions in $L^{\perp}$ . The first marginal of the limit plan is supported on $L$ . By (5.6), we have

(5.36)

D\tilde{v}_{\infty}(y)\in L\qquad\text{for a.e. }y\in\Omega_{\infty}^{*}.

By convexity, for any $y\in\Omega_{\infty}^{*},p\in\partial\tilde{v}_{\infty}(y)$ and $e\in L^{\perp}$ , one has $p\cdot e=0.$ Hence $\tilde{v}_{\infty}$ is constant on every convex fiber $\Omega^{*}_{\infty,\xi}:=\Omega_{\infty}^{*}\cap(\xi+L^{\perp})$ . There is a finite convex function $\varphi$ on the open convex projected domain $\Omega_{L}^{*}=P_{L}\Omega_{\infty}^{*}$ such that

(5.37)

\tilde{v}_{\infty}(y)=\varphi(P_{L}y),\qquad y\in\Omega_{\infty}^{*}.

Define

\phi(x):=\sup_{\xi\in\Omega_{L}^{*}}\{x\cdot\xi-\varphi(\xi)\},\qquad x\in L.

For $x\in L$ , (5.2) and the preceding factorization give

	$\displaystyle\tilde{u}_{\infty}(x)$	$\displaystyle=\sup_{y\in\Omega_{\infty}^{*}}\{x\cdot y-\tilde{v}_{\infty}(y)\}$
		$\displaystyle=\sup_{y\in\Omega_{\infty}^{*}}\{x\cdot P_{L}y-\varphi(P_{L}y)\}$
		$\displaystyle=\sup_{\xi\in\Omega_{L}^{*}}\{x\cdot\xi-\varphi(\xi)\}=\phi(x).$

Thus both identities in (5.26) hold.

We next integrate along the $L^{\perp}$ -fibers. Put $q=n-m$ and identify $L$ with $\mathbb{R}^{m}$ . Fubini gives

(P_{L})_{\#}\bigl(\rho_{\infty}^{*}\mathbf{1}_{\Omega_{\infty}^{*}}\,\,\mathrm{d}y\bigr)=\rho_{L}^{*}(\xi)\,\,\mathrm{d}\xi,\qquad\rho_{L}^{*}(\xi)=\rho_{\infty}^{*}\mathcal{H}^{q}(\Omega^{*}_{\infty,\xi})=\rho_{\infty}^{*}\mathcal{H}^{q}\bigl(\Omega_{\infty}^{*}\cap(\xi+L^{\perp})\bigr).

For $\xi_{0},\xi_{1}\in\Omega_{L}^{*}$ and $0<t<1$ , convexity gives

(5.38)

(1-t)\Omega^{*}_{\infty,\xi_{0}}+t\Omega^{*}_{\infty,\xi_{1}}\subset\Omega^{*}_{\infty,(1-t)\xi_{0}+t\xi_{1}}.

The $q$ -dimensional Brunn–Minkowski inequality therefore shows that $(\rho_{L}^{*})^{1/q}$ is concave wherever it is finite.

It remains to exclude infinite fibers without assuming that $\Omega_{\infty}^{*}$ is bounded. If $Q\Subset\Omega_{L}^{*}$ , by local boundedness of $\partial\varphi$ , we have

\sup\{|\xi\cdot x|:\xi\in Q,\ x\in\partial\varphi(\xi)\}<\infty.

By (5.26), almost every point of $\Omega_{\infty}^{*}\cap P_{L}^{-1}(Q)$ lies in $\{y\in\Omega_{\infty}^{*}:y\cdot D\tilde{v}_{\infty}(y)<R^{2}\}$ for some fixed $R$ . That set has finite limiting mass, and hence

(5.39)

\int_{Q}\rho_{L}^{*}\,d\xi<\infty.

Thus $\rho_{L}^{*}<\infty$ almost everywhere. Suppose that $\rho_{L}^{*}(\xi_{0})=\infty$ at an interior point. Fix a ball $B\Subset\Omega_{L}^{*}$ not containing $\xi_{0}$ . For every fixed $t\in(0,1)$ , by (5.38), the fiber over $(1-t)\xi_{0}+t\eta$ has infinite volume for every $\eta\in B$ . These base points form a set of positive measure, which contradicts (5.39). Therefore every fiber over $\xi\in\Omega_{L}^{*}$ is bounded and

0<\rho_{L}^{*}(\xi)<\infty\qquad(\xi\in\Omega_{L}^{*}).

In particular $(\rho_{L}^{*})^{1/q}$ is a finite concave function on $\Omega_{L}^{*}$ .

Finally, we identify the projected transport plan and preserve the mass identity. Let $\widehat{\pi}$ be the image of the limiting transport plan under $(x,y)\mapsto(x,P_{L}y)$ . Its marginals are $\rho\mathbf{1}_{\Omega_{\infty}}\,d\mathcal{H}^{m}\lfloor L$ and $\rho_{L}^{*}\,d\xi$ . By (5.6) and (5.26), it is concentrated on $x\in\partial\varphi(\xi)$ . Since $\rho_{L}^{*}\,d\xi$ is absolutely continuous and $\varphi$ is differentiable almost everywhere,

(5.40)

\widehat{\pi}=(D\varphi,\operatorname{id})_{\#}(\rho_{L}^{*}\,\,\mathrm{d}\xi),\qquad(D\varphi)_{\#}(\rho^{*}_{L}\,\,\mathrm{d}\xi)=\rho\mathbf{1}_{\Omega_{\infty}}\,\,\mathrm{d}\mathcal{H}^{m}\lfloor L.

Since the first marginal of $\pi$ is supported on $L$ ,

x\cdot P_{L}y=x\cdot y\qquad\text{for $\pi$-a.e. }(x,y).

Consequently, by Lemma 5.1(iii), we have for every $r>0$ ,

\widehat{\pi}\{(x,\xi):x\cdot\xi<r^{2}\}=r^{n}.

Using (5.40), we obtain (5.27). Thus the mass identity is preserved. ∎

6. Rigidity of normalized limits

We prove the equality case for the weighted transports obtained in Section 5, and then lift the resulting homogeneity to the original normalized limit. The concavity of $\rho^{1/q}$ first yields a radial inequality with a sharp equality condition.

Lemma 6.1.

Let $q>0$ . Let $C$ be convex with $0\in\overline{C}$ , and suppose that $\rho^{1/q}$ is finite, nonnegative, and concave on $C$ . Then

(6.1)

x\cdot D\rho(x)\leq q\rho(x)

at every differentiability point. On every admissible ray, the function $r\mapsto r^{-q}\rho(r\theta)$ is nonincreasing. It is constant on an interval precisely when $\rho(tx)=t^{q}\rho(x)$ there.

Proof.

Set $w=\rho^{1/q}$ . Fix an admissible ray and $0<\delta<s<r$ . Concavity and nonnegativity imply

w(s\theta)\geq\frac{r-s}{r-\delta}w(\delta\theta)+\frac{s-\delta}{r-\delta}w(r\theta)\geq\frac{s-\delta}{r-\delta}w(r\theta).

Letting $\delta\rightarrow 0$ , we have $w(s\theta)/s\geq w(r\theta)/r$ . Hence $r^{-q}\rho(r\theta)$ is nonincreasing, and differentiation proves (6.1). Now, $r^{-q}\rho(r\theta)$ is locally absolutely continuous, hence, equality in (6.1) almost everywhere implies that $r^{-q}\rho(r\theta)$ is a constant, which implies $\rho(tx)=t^{q}\rho(x)$ . ∎

The limiting transports below may have infinite total mass. We shall use Proposition A.1, proved in Appendix A, which implies strict convexity, local $C^{1}$ regularity, and the homeomorphism property for such locally finite transports. Hölder continuity under the assumptions above follows from the interior regularity theory for the Monge–Ampère equation.

Let $m\geq 1$ . Let $C,C^{*}\subset\mathbb{R}^{m}$ be open convex domains and let $\phi$ on $C$ and $\varphi$ on $C^{*}$ be the corresponding convex potential functions whose extensions satisfy (2.1) and (2.2) in Section 2. Since $\phi$ and $\varphi$ are Legendre duals of each other within the convex domains $C$ and $C^{*}$ , we can express this minimal convex extension in the following equivalent form.

(6.2)		$\displaystyle\underline{\phi}(x)$	$\displaystyle=\sup\{\ell(x):\ell\text{ is affine},\ \ell\leq\phi\text{ in }C,\ D\ell\in\overline{C^{*}}\},$
(6.2)		$\displaystyle\underline{\varphi}(y)$	$\displaystyle=\sup\{\ell(y):\ell\text{ is affine},\ \ell\leq\varphi\text{ in }C^{*},\ D\ell\in\overline{C}\}.$

For simplicity of notation, we continue to denote the extended functions by $\phi$ and $\varphi$ .

Lemma 6.2.

Let $m\geq 1$ and $q\geq 0$ . Let $C,C^{*},\phi$ and $\varphi$ be given as above. Assume that $D\phi$ transports $\rho\,\,\mathrm{d}x$ to $\rho^{*}\,\,\mathrm{d}y$ . If $q>0$ , assume that $\rho^{1/q}$ and $(\rho^{*})^{1/q}$ are positive and concave in $C$ and $C^{*}$ , respectively. If $q=0$ , assume that $\rho,\rho^{*}$ are positive constants. Suppose also that $(x_{c},y_{c})\in\overline{C}\times\overline{C^{*}}$ satisfies

y_{c}\in\partial\phi(x_{c}),\qquad x_{c}\in\partial\varphi(y_{c}),\qquad\phi(x_{c})+\varphi(y_{c})=x_{c}\cdot y_{c},

and, for every $h>0$ , the two sets

\bigl\{x\in C:(x-x_{c})\cdot(D\phi(x)-y_{c})<h\bigr\},\qquad\bigl\{y\in C^{*}:(y-y_{c})\cdot(D\varphi(y)-x_{c})<h\bigr\}

are bounded and have finite mass. Then

\phi,\varphi\ \hbox{are strictly convex},\qquad\phi\in C^{1,\alpha}_{\rm loc}(C),\quad\varphi\in C^{1,\alpha}_{\rm loc}(C^{*})

with a possibly compact-set-dependent exponent $\alpha>0$ , and the following facts hold.

(i)

The two measures are doubling. More precisely, if $E$ is a bounded ellipsoid centered at a point of $\overline{C}$ , or respectively of $\overline{C^{*}}$ , then

(6.3)

\int_{E}\rho\,\,\mathrm{d}x\leq 2^{m+q}\int_{\frac{1}{2}E}\rho\,\,\mathrm{d}x,\qquad\int_{E}\rho^{*}\,\,\mathrm{d}y\leq 2^{m+q}\int_{\frac{1}{2}E}\rho^{*}\,\,\mathrm{d}y.

When $q=0$ , the exponent on the right is $m$ . In these inequalities $\rho$ and $\rho^{*}$ are extended by zero outside $C$ and $C^{*}$ , respectively.

(ii)

The maps

(6.4) $T:=D\phi:C\longrightarrow C^{*},\qquad S:=D\varphi:C^{*}\longrightarrow C$

are mutually inverse homeomorphisms. In particular,

$K\Subset C\Longrightarrow T(K)\Subset C^{*},\qquad K^{*}\Subset C^{*}\Longrightarrow S(K^{*})\Subset C.$
(iii)

After possibly decreasing the local exponent,

$\phi\in C^{2,\alpha^{\prime}}_{\rm loc}(C),\qquad\varphi\in C^{2,\alpha^{\prime}}_{\rm loc}(C^{*}).$

At corresponding points $y=T(x)$ ,

(6.5) $H:=D^{2}\phi(x)>0,\qquad D^{2}\varphi(y)=H^{-1}.$
(iv)

After translating $x_{c},y_{c}$ to the origin, denote

$\Psi(x)=x\cdot T(x),\qquad\Psi^{*}(y)=y\cdot S(y),\qquad M(h)=\int_{\{\Psi<h\}}\rho.$

Then $\Psi_{\#}(\rho\,\,\mathrm{d}x)$ is absolutely continuous on compact subintervals of $(0,\infty)$ , $M$ is locally absolutely continuous, and

$M^{\prime}(h)=\int_{\{\Psi=h\}}\frac{\rho}{|\nabla\Psi|}\,d\mathcal{H}^{m-1}\quad\text{for a.e. }h>0.$

Proof.

Assume first that $q>0$ and write $w=(\rho)^{1/q}$ . If $E$ is centered at $z\in\overline{C}$ , the map $x\mapsto z+(x-z)/2$ sends $E\cap C$ into $\frac{1}{2}E\cap C$ . If $z\in C$ , by concavity and nonnegativity, we have

w\left(\frac{x+z}{2}\right)\geq\frac{w(z)+w(x)}{2}\geq\frac{w(x)}{2}.

When $z\in\partial C$ , choose $z_{j}\in C$ with $z_{j}\to z$ , apply the same inequality at $(x+z_{j})/2$ , and pass to the limit using continuity of $w$ at the interior point $(x+z)/2$ . This implies $w((x+z)/2)\geq w(x)/2$ without assuming a boundary value for $w$ . Consequently,

\int_{\frac{1}{2}E}\rho\geq 2^{-m}\int_{E\cap C}\rho((x+z)/2)\,\,\mathrm{d}x\geq 2^{-(m+q)}\int_{E}\rho.

The proof of the equality for $\rho^{*}$ is the same. For $q=0$ , the weights are constant and the same homothety gives the factor $2^{m}$ . This proves (6.3). The homeomorphism and regularity assertions (6.4) and (6.5) follow from Proposition A.1 and Caffarelli’s standard interior regularity theory.

We next justify the coarea formula and absolute continuity. At every interior point,

\nabla\Psi=T+Hx,\qquad x\cdot\nabla\Psi=\Psi+x^{t}Hx\geq\Psi.

Fix $0<h_{0}<h_{1}<h_{2}<\infty$ . By hypothesis, $\{\Psi<h_{2}\}\subset B_{R}$ for some $R<\infty$ . Hence

(6.6)

|\nabla\Psi|\geq\frac{h_{0}}{R}\qquad\text{on }\{h_{0}\leq\Psi\leq h_{1}\}.

Choose a sequence $\chi_{\ell}\in C_{c}^{\infty}(C)$ such that

0\leq\chi_{\ell}\uparrow 1\qquad\text{pointwise in }C.

If $N\subset[h_{0},h_{1}]$ is null, by coarea and (6.6), we have

\int_{\Psi^{-1}(N)}\rho\chi_{\ell}\,\,\mathrm{d}x\leq\frac{R}{h_{0}}\int_{N}\int_{\{\Psi=h\}}\rho\chi_{\ell}\,d\mathcal{H}^{m-1}\,dh=0.

Monotone convergence shows that $\Psi_{\#}(\rho\,\,\mathrm{d}x)$ is absolutely continuous on $[h_{0},h_{1}]$ . Applying coarea once more gives, for every Borel $E\subset[h_{0},h_{1}]$ ,

\int_{\{\Psi\in E\}}\rho\,\,\mathrm{d}x=\int_{E}\int_{\{\Psi=h\}}\frac{\rho}{|\nabla\Psi|}\,d\mathcal{H}^{m-1}\,dh.

Since the band is arbitrary, $M$ is locally absolutely continuous on $(0,\infty)$ , and

M^{\prime}(h)=\int_{\{\Psi=h\}}\frac{\rho}{|\nabla\Psi|}\,d\mathcal{H}^{m-1}\qquad\text{for a.e. }h>0.

The same argument applies to $\Psi^{*}$ . ∎

The proof of the next theorem is modeled on the monotonicity formula and its equality case in Collins–Tong [12, Theorem 3.1], using the calculation in [12, proof of Proposition 3.1]. We include the details because neither homogeneity of the densities nor conic structure of the domains is assumed here.

Theorem 6.3.

Let $m\geq 1$ , $q\geq 0$ , and put $n=m+q$ . Let $\rho\,\,\mathrm{d}x$ and $\rho^{*}\,\,\mathrm{d}y$ be locally finite measures on convex domains $C,C^{*}\subset\mathbb{R}^{m}$ . If $q>0$ , assume that $\rho^{1/q}$ and $(\rho^{*})^{1/q}$ are nonnegative and concave. If $q=0$ , assume that $\rho,\rho^{*}$ are positive constants. Let $\phi,\varphi$ be dual convex potentials satisfying (6.2), and assume that $D\phi$ transports $\rho\,\,\mathrm{d}x$ to $\rho^{*}\,\,\mathrm{d}y$ . Assume that $C$ and $C^{*}$ are the interiors of the supports of $\rho,\rho^{*},$ respectively, and $0\in\overline{C}\cap\overline{C^{*}}$ . Assume also that

(6.7)

\phi(0)+\varphi(0)=0,\qquad 0\in\partial\phi(0),\qquad 0\in\partial\varphi(0).

Suppose that, for every $h>0$ , the sets

\{x\in C:x\cdot D\phi(x)<h\},\qquad\{y\in C^{*}:y\cdot D\varphi(y)<h\}

are bounded and have finite mass. If

r^{-n}\int_{\{x\cdot D\phi(x)<r^{2}\}}\rho(x)\,\,\mathrm{d}x

is a positive constant for $0<r<\infty$ , then $C,C^{*}$ are cones, $\rho,\rho^{*}$ are homogeneous of degree $q$ , and the normalized potentials $\phi-\phi(0),\ \varphi-\varphi(0)$ are homogeneous of degree two.

Proof.

After replacing $\phi$ by $\phi-\phi(0)$ and $\varphi$ by $\varphi-\varphi(0)$ , we may assume that

(6.8)

\phi(0)=\varphi(0)=0,\ \ 0\in\partial\phi(0),\quad 0\in\partial\varphi(0).

Set

T=D\phi,\qquad S=D\varphi,\qquad\Psi(x)=x\cdot T(x),\qquad\Psi^{*}(y)=y\cdot S(y),

and

K_{h}=\{x\in C:\Psi(x)<h\},\qquad L_{h}=\{y\in C^{*}:\Psi^{*}(y)<h\},\qquad M(h)=\int_{K_{h}}\rho.

By Lemma 6.2,

T(K_{h})=L_{h},\qquad M(h)=\int_{L_{h}}\rho^{*}(y)\,\,\mathrm{d}y,

and all the regularity and coarea formulas used below hold. The mass assumption is exactly

(6.9)

M(h)=ch^{n/2}\qquad(h>0).

For almost every $h>0$ , define

A_{C}(h)=\int_{\{\Psi=h\}}\rho\frac{x\cdot\nabla\Psi}{|\nabla\Psi|}\,\,\mathrm{d}\mathcal{H}^{m-1}

and define $A_{C^{*}}(h)$ analogously.

Step 1: estimates for $A_{C}$ and $A_{C^{*}}$ . We claim that

(6.10)

A_{C}(h)\leq nM(h),\qquad A_{C^{*}}(h)\leq nM(h).

It suffices to prove the first inequality. For $\theta\in\mathbb{S}^{m-1}$ , write

C\cap\mathbb{R}_{+}\theta=(0,R_{C}(\theta))\theta,\qquad K_{h}\cap\mathbb{R}_{+}\theta=(0,R_{h}(\theta))\theta,

and let

\Gamma_{C}:=\{\theta\in\mathbb{S}^{m-1}:R_{C}(\theta)>0\}.

Convexity and (6.8) imply that

r\longmapsto\Psi(r\theta)=r\,\partial_{r}\phi(r\theta)

is nondecreasing. By Lemma 6.1, $\rho_{\theta}(r):=r^{-q}\rho(r\theta)$ is nonincreasing. Since $\partial_{r}\Psi>0$ on every positive level set, $\{\Psi=h\}$ is the radial graph $x=R_{h}(\theta)\theta$ over the set $\{\theta:R_{h}(\theta)<R_{C}(\theta)\}$ . The area formula for radial graphs gives

A_{C}(h)=\int_{\{\theta:R_{h}(\theta)<R_{C}(\theta)\}}\rho\bigl(R_{h}(\theta)\theta\bigr)R_{h}(\theta)^{m}\,\,\mathrm{d}\theta.

Hence

	$\displaystyle nM(h)-A_{C}(h)$	$\displaystyle=n\int_{\{R_{h}<R_{C}\}}\int_{0}^{R_{h}}\bigl(\rho_{\theta}(r)-\rho_{\theta}(R_{h})\bigr)r^{n-1}\,\,\mathrm{d}r\,\,\mathrm{d}\theta$
		$\displaystyle\quad+n\int_{\{R_{h}=R_{C}\}}\int_{0}^{R_{C}}\rho_{\theta}(r)r^{n-1}\,\,\mathrm{d}r\,\,\mathrm{d}\theta.$

This proves (6.10). Since $\rho>0$ on $C$ , equality holds only if the set of directions for which $R_{h}=R_{C}$ has spherical measure zero and $\rho_{\theta}$ is constant on $(0,R_{h})$ for almost every remaining direction.

Step 2: equality in the two estimates. Let $y=T(x)$ and $H=D^{2}\phi(x)$ . Then

D^{2}\phi(x)>0,\qquad D^{2}\varphi(y)=\left(D^{2}\phi(x)\right)^{-1},\qquad\Psi^{*}(T(x))=\Psi(x),

and

\nabla\Psi=T+D^{2}\phi(x)x,\qquad\nabla\Psi^{*}(T(x))=x+\left(D^{2}\phi(x)\right)^{-1}T(x).

Fix $0\leq\eta\in C_{c}((0,\infty))$ . Choose increasing sequences

\alpha_{\ell}\in C_{c}^{\infty}(C),\qquad\beta_{\ell}\in C_{c}^{\infty}(C^{*}),\qquad 0\leq\alpha_{\ell}\rightarrow 1,\quad 0\leq\beta_{\ell}\rightarrow 1,

and set

\gamma_{\ell}(x)=\alpha_{\ell}(x)\beta_{\ell}(T(x)),\qquad\gamma_{\ell}^{*}(y)=\alpha_{\ell}(S(y))\beta_{\ell}(y).

Then $\gamma_{\ell}^{*}(T(x))=\gamma_{\ell}(x).$ By coarea, the transport identity, and monotone convergence, we have

	$\displaystyle\int_{0}^{\infty}\eta(h)A_{C^{*}}(h)\,\,\mathrm{d}h$	$\displaystyle=\lim_{\ell\to\infty}\int_{C^{}}\eta(\Psi^{})\,\rho^{}\,\gamma_{\ell}^{}\,y\cdot\nabla\Psi^{*}\,\,\mathrm{d}y$
		$\displaystyle=\lim_{\ell\to\infty}\int_{C}\eta(\Psi)\,\rho\,\gamma_{\ell}\,\bigl(\Psi+T^{t}\left(D^{2}\phi(x)\right)^{-1}T\bigr)\,\,\mathrm{d}x$
		$\displaystyle=\lim_{\ell\to\infty}\int_{0}^{\infty}\eta(h)\int_{\{\Psi=h\}}\frac{\rho\gamma_{\ell}}{\|\nabla\Psi\|}\bigl(h+T^{t}\left(D^{2}\phi(x)\right)^{-1}T\bigr)\,\,\mathrm{d}\mathcal{H}^{m-1}\,\,\mathrm{d}h$
		$\displaystyle=\int_{0}^{\infty}\eta(h)\int_{\{\Psi=h\}}\frac{\rho}{\|\nabla\Psi\|}\bigl(h+T^{t}\left(D^{2}\phi(x)\right)^{-1}T\bigr)\,\,\mathrm{d}\mathcal{H}^{m-1}\,\,\mathrm{d}h.$

Since $\eta$ is arbitrary, for almost every $h>0$ ,

	$\displaystyle A_{C}(h)$	$\displaystyle=\int_{\{\Psi=h\}}\frac{\rho}{\|\nabla\Psi\|}\bigl(h+x^{t}D^{2}\phi(x)x\bigr)\,\,\mathrm{d}\mathcal{H}^{m-1},$
	$\displaystyle A_{C^{*}}(h)$	$\displaystyle=\int_{\{\Psi=h\}}\frac{\rho}{\|\nabla\Psi\|}\bigl(h+T^{t}\left(D^{2}\phi(x)\right)^{-1}T\bigr)\,\,\mathrm{d}\mathcal{H}^{m-1}.$

Since $h=\Psi=x\cdot T$ on $\{\Psi=h\}$ ,

	$\displaystyle A_{C}(h)+A_{C^{*}}(h)-4hM^{\prime}(h)$	$\displaystyle=\int_{\{\Psi=h\}}\frac{\rho}{\|\nabla\Psi\|}\left\|\left(D^{2}\phi(x)\right)^{1/2}x-\left(D^{2}\phi(x)\right)^{-1/2}T\right\|^{2}\,\,\mathrm{d}\mathcal{H}^{m-1}$
		$\displaystyle=:\mathcal{E}(h)\geq 0.$

By (6.9), $4hM^{\prime}(h)=2nM(h).$ Combining this identity with (6.10) gives

2nM(h)\geq A_{C}(h)+A_{C^{*}}(h)=2nM(h)+\mathcal{E}(h)\geq 2nM(h).

Therefore

(6.11)

A_{C}(h)=A_{C^{*}}(h)=nM(h),\qquad\mathcal{E}(h)=0\quad\text{for a.e. }h>0.

Step 3: homogeneity of the densities and potentials. Choose $h_{j}\rightarrow\infty$ for which (6.11) holds. Outside a fixed null set of directions, the equality case in Step 1 gives

R_{h_{j}}<R_{C}\quad\text{and}\quad\rho_{\theta}\ \text{constant on }(0,R_{h_{j}}(\theta))

for every $j$ . Since $K_{h_{j}}\rightarrow C$ , $R_{h_{j}}(\theta)\rightarrow R_{C}(\theta).$ It follows that

(6.12)

\rho(tx)=t^{q}\rho(x),\qquad\rho^{*}(ty)=t^{q}\rho^{*}(y),

whenever $x,tx\in C$ and $y,ty\in C^{*}$ .

By strict convexity and (6.8), we have

\Psi>0\quad\text{in }C\setminus\{0\}.

For every $I\Subset(0,\infty)$ , by coarea formula and (6.11), we have

0=\int_{I}\mathcal{E}(h)\,\,\mathrm{d}h=\int_{\{\Psi\in I\}}\rho(x)\left|\left(D^{2}\phi(x)\right)^{1/2}x-\left(D^{2}\phi(x)\right)^{-1/2}T(x)\right|^{2}\,\,\mathrm{d}x.

Exhausting $\{\Psi>0\}$ and using continuity gives $D^{2}\phi(x)x=D\phi(x)\ (x\in C).$ Consequently,

D\bigl(x\cdot D\phi-2\phi\bigr)=0,

so $x\cdot D\phi(x)-2\phi(x)=c_{0}\ (x\in C)$ for some constant $c_{0}$ . Fix $x\in C$ and set $w(t)=\phi(tx)$ . On every interval on which $tx\in C$ ,

tw^{\prime}(t)-2w(t)=c_{0},

and hence $w(t)=At^{2}-\frac{c_{0}}{2}.$ By convexity and (6.8),

0\leq\phi(tx)\leq t\phi(x)\longrightarrow 0\qquad\text{as }t\rightarrow 0.

Thus $c_{0}=0$ , and

(6.13)

\phi(tx)=t^{2}\phi(x)

whenever $x,tx\in C$ . The same argument applies to $\varphi$ .

Step 4: the domains are cones. Recall that

K_{h}=\{x\in C:\Psi(x)<h\}.

Fix $r>1$ . Since $C$ is convex and $0\in\overline{C}$ , we have

r^{-1}C\subset C.

The homogeneity identities already proved imply

K_{r^{2}}=r\bigl(K_{1}\cap r^{-1}C\bigr).

Indeed, if $z\in K_{1}\cap r^{-1}C$ , then $rz\in C$ and

\Psi(rz)=r^{2}\Psi(z)<r^{2}.

The converse follows by applying the same identity to $z=x/r$ .

Using $\rho(rz)=r^{q}\rho(z)$ and $n=m+q$ , we obtain

	$\displaystyle M(r^{2})$	$\displaystyle=\int_{K_{r^{2}}}\rho(x)\,\,\mathrm{d}x$
		$\displaystyle=r^{m}\int_{K_{1}\cap r^{-1}C}\rho(rz)\,\,\mathrm{d}z$
		$\displaystyle=r^{n}\int_{K_{1}\cap r^{-1}C}\rho(z)\,\,\mathrm{d}z.$

Consequently,

(6.14)

r^{-n}M(r^{2})=\int_{K_{1}\cap r^{-1}C}\rho(z)\,\,\mathrm{d}z.

By (6.9), the left-hand side of (6.14) equals

M(1)=\int_{K_{1}}\rho(z)\,\,\mathrm{d}z.

Since $K_{1}\cap r^{-1}C\subset K_{1}$ and $\rho>0$ in $C$ , it follows that

K_{1}\subset r^{-1}C.

Equivalently,

(6.15)

rK_{1}\subset C\qquad(r>1).

Now fix $x\in C$ and $t>1$ . Since $0\in\overline{C}$ , we have $sx\in C$ for every $0<s<1$ . Moreover,

\Psi(sx)=s^{2}\Psi(x)\longrightarrow 0\qquad(s\rightarrow 0).

Thus we may choose $s\in(0,1)$ such that $sx\in K_{1}$ . Applying (6.15) with $r=t/s>1$ , we obtain

tx=\frac{t}{s}(sx)\in C.

For $0<t<1$ , the inclusion $tx\in C$ follows directly from the convexity of $C$ and $0\in\overline{C}$ . Hence

tC=C\qquad(t>0),

so $C$ is a cone. The same argument applies to $C^{*}$ . The homogeneity identities for $\rho,\rho^{*},\phi,\varphi$ are therefore global. ∎

We now identify the normalized limit.

Theorem 6.4.

Under (4.3)–(4.6), the centered sub-level sets at heights $1$ and $1/4$ subconverge, and their limits satisfy

S_{1/4}^{c}[\tilde{u}_{\infty}](0)=\tfrac{1}{2}S_{1}^{c}[\tilde{u}_{\infty}](0).

Proof.

Apply Lemma 5.1. After passing to a subsequence, we obtain a locally uniform limit $\tilde{u}_{\infty}$ , its conjugate $\tilde{v}_{\infty}=\tilde{u}_{\infty}^{*}$ , the compatible limiting plan $\pi$ , and the Hausdorff limits of the two centered sub-level sets. Let $\mu$ be the first marginal of $\pi$ , and set

L=\operatorname{aff}(\operatorname{spt}\mu),\qquad m=\dim L.

By Lemma 5.3, $L$ is a linear subspace containing the origin and $1\leq m\leq n$ .

Suppose first that $m=n$ . Then $\mu=\rho_{\infty}\mathbf{1}_{C}\,\,\mathrm{d}x$ for an open convex set $C$ and a constant $\rho_{\infty}>0$ . The argument in the proof of Lemma 5.4, from (5.29) through the subsequent identification of the full second marginal, does not use $m<n$ . Therefore, we have

(\operatorname{pr}_{2})_{\#}\pi=\rho^{*}_{\infty}\mathbf{1}_{C^{*}}\,\,\mathrm{d}y,\qquad C^{*}=\operatorname{int}(\operatorname{dom}\tilde{v}_{\infty}),

for some constant $\rho^{*}_{\infty}>0$ . Let $\phi=\tilde{u}_{\infty},\varphi=(\tilde{u}_{\infty}+I_{C})^{*}$ . For $\pi$ -almost every $(x,y)$ , we have $x\in C$ , $y\in C^{*}$ , and

\tilde{u}_{\infty}(x)+\tilde{v}_{\infty}(y)=x\cdot y.

Consequently,

\varphi(y)\geq x\cdot y-\tilde{u}_{\infty}(x)=\tilde{v}_{\infty}(y).

Since $\varphi\leq\tilde{v}_{\infty}$ by definition, the density of the second projection of $\pi$ and continuity imply $\varphi=\tilde{v}_{\infty}$ on $C^{*}$ . Hence by (5.2) and the definition of $\varphi$ , we have

\phi(x)=\sup_{y\in C^{*}}\{x\cdot y-\varphi(y)\},\qquad\varphi(y)=\sup_{x\in C}\{x\cdot y-\phi(x)\}.

Thus $\phi$ and $\varphi$ are Legendre dual relative to $C$ and $C^{*}$ . Proposition A.1, applied in both directions, shows that

D\phi:C\longrightarrow C^{*},\qquad D\varphi:C^{*}\longrightarrow C

are homeomorphisms. Lemma 5.1(iii) and Proposition 4.1 show that the projections of

\operatorname{spt}\bigl(\pi\lfloor\{(x,y):x\cdot y<r^{2}\}\bigr)

onto the two factors are bounded. Since $D\phi,D\varphi$ are continuous and the limiting densities are positive constants, it follows that, for every $r>0$ , the sets

\{x\in C:x\cdot D\phi(x)<r^{2}\},\qquad\{y\in C^{*}:y\cdot D\varphi(y)<r^{2}\}

are bounded. Moreover, Lemma 5.1(iii) gives

\mu\{x\in C:x\cdot D\phi(x)<r^{2}\}=\pi\{(x,y):x\cdot y<r^{2}\}=r^{n},

and hence

r^{-n}\mu\{x\in C:x\cdot D\phi(x)<r^{2}\}=1\qquad(r>0).

Thus the limit is precisely the constant-density equality case of the Collins–Tong monotonicity formula. Since $\phi(0)=\varphi(0)=0$ and $\phi,\varphi\geq 0$ , the normalization gives (6.7). Hence all the hypotheses of Theorem 6.3 with $q=0$ are satisfied. Its $q=0$ argument is precisely the constant-density rigidity of Collins–Tong [12, Theorem 3.1], as used in [12, proof of Theorem 4.1]. Hence $C,C^{*}$ are cones and $\phi=\tilde{u}_{\infty},\varphi$ are 2-homogeneous.

It remains to consider the case $m<n$ . By Lemma 5.3, we have $1\leq m<n$ and, for every fixed $R<\infty$ ,

\sup_{x\in D_{R}(\tilde{u}_{j},0)}\operatorname{dist}(x,L)\longrightarrow 0.

Hence the hypotheses of Lemma 5.4 are satisfied.

Put $q=n-m$ . Lemma 5.4 gives a full-dimensional dual domain $\Omega_{\infty}^{*}$ , dual convex potentials $\phi$ on $C$ and $\varphi$ on $C^{*}:=P_{L}\Omega_{\infty}^{*}$ , and a constant $\rho^{*}_{\infty}>0$ such that

(6.16)

\tilde{v}_{\infty}(y)=\varphi(P_{L}y),\qquad y\in\Omega_{\infty}^{*}.

Moreover, with

\rho^{*}_{L}(\xi)=\rho^{*}_{\infty}\,\mathcal{H}^{q}(\Omega_{\infty}^{*}\cap(\xi+L^{\perp})),

every fiber has finite positive volume, $(\rho^{*}_{L})^{1/q}$ is positive and concave, and

(6.17)

(D\varphi)_{\#}(\rho^{*}_{L}\,\,\mathrm{d}\xi)=\rho\mathbf{1}_{C}\,d\mathcal{H}^{m}\lfloor L,\qquad r^{-n}\int_{\{\xi\cdot D\varphi(\xi)<r^{2}\}}\rho^{*}_{L}(\xi)\,\,\mathrm{d}\xi=1\quad(r>0).

By (5.2) and (6.16),

(6.18)

\phi(x)=\tilde{u}_{\infty}(x)=\sup_{y\in\Omega_{\infty}^{*}}\{x\cdot y-\tilde{v}_{\infty}(y)\}=\sup_{\xi\in C^{*}}\{x\cdot\xi-\varphi(\xi)\},\qquad x\in L.

Define

\underline{\phi}:=(\varphi+I_{C^{*}})^{*},\qquad\underline{\varphi}:=(\phi+I_{C})^{*}.

By (6.18),

\underline{\phi}=\phi\quad\text{on }L,\qquad x\cdot\xi-\phi(x)\leq\varphi(\xi)\quad(x\in C,\ \xi\in C^{*}).

Hence

\underline{\varphi}(\xi)=\sup_{x\in C}\{x\cdot\xi-\phi(x)\}\leq\varphi(\xi)\qquad(\xi\in C^{*}).

On the other hand, for $\rho_{L}^{*}\,\,\mathrm{d}\xi$ -almost every $\xi\in C^{*}$ , (6.17) implies $x=D\varphi(\xi)\in C$ . By Legendre duality we have

\phi(x)+\varphi(\xi)=x\cdot\xi,

and therefore

\underline{\varphi}(\xi)\geq x\cdot\xi-\phi(x)=\varphi(\xi).

Thus $\underline{\varphi}=\varphi$ almost everywhere in $C^{*}$ . Since both functions are finite and convex in $C^{*}$ , they are continuous, and hence

\underline{\varphi}=\varphi\qquad\text{in }C^{*}.

Therefore $\underline{\phi}$ and $\underline{\varphi}$ are the minimal convex extensions of $\phi$ and $\varphi$ , respectively, and satisfy (6.2).

Moreover,

\underline{\phi}^{*}=\operatorname{cl}(\varphi+I_{C^{*}}),\qquad\underline{\varphi}^{*}=\operatorname{cl}(\phi+I_{C}).

Consequently,

\partial\underline{\phi}(\mathbb{R}^{m})\subset\overline{C^{*}},\qquad\partial\underline{\varphi}(\mathbb{R}^{m})\subset\overline{C}.

The normalization and the continuity of $\tilde{u}_{\infty}$ provide a sequence $x_{\ell}\in C$ such that

x_{\ell}\to 0,\qquad\phi(x_{\ell})\to 0.

Moreover, (5.2) and $\tilde{v}_{\infty}(0)=0$ provide $y_{\ell}\in\Omega^{*}_{\infty}$ such that

y_{\ell}\to 0,\qquad\tilde{v}_{\infty}(y_{\ell})\to 0.

Setting $\xi_{\ell}=P_{L}y_{\ell}$ , (6.16) gives

\xi_{\ell}\to 0,\qquad\varphi(\xi_{\ell})\to 0.

Consequently the two minimal extensions are nonnegative, vanish at the origin, and satisfy

\underline{\phi}(0)=\underline{\varphi}(0)=0,\qquad 0\in\partial\underline{\phi}(0),\qquad 0\in\partial\underline{\varphi}(0),

so (6.7) holds.

Since $\rho^{1/q}$ and $(\rho^{*}_{L})^{1/q}$ are concave, by the proof of (6.3) in Lemma 6.2 we have that $\rho\,\mathrm{d}\mathcal{H}^{m}$ and $\rho^{*}_{L}\,\mathrm{d}\xi$ are locally doubling. Proposition A.1 shows that $D\phi$ and $D\varphi$ are homeomorphisms. Let

\widehat{\pi}:=(\operatorname{id},P_{L})_{\#}\pi,\qquad\widehat{G}_{r}:=\{(x,\xi)\in L\times L:x\cdot\xi<r^{2}\}.

Since the first marginal of $\pi$ is supported on $L$ , we have $x\cdot P_{L}y=x\cdot y$ for $\pi$ -almost every $(x,y)$ , and hence

\widehat{\pi}\lfloor\widehat{G}_{r}=(\operatorname{id},P_{L})_{\#}(\pi\lfloor G_{r}).

For fixed $r>0$ , Proposition 4.1 implies the supports of $\pi_{j}\lfloor G_{r}$ in a common compact set. Passing to the limit in (5.5) shows that $\pi\lfloor G_{r}$ is supported in the same compact set. Therefore,

\widehat{\pi}\lfloor\widehat{G}_{r}=(\operatorname{id},P_{L})_{\#}(\pi\lfloor G_{r})

has compact support.

Set

A_{r}:=\{x\in C:x\cdot D\phi(x)<r^{2}\},\qquad A^{\prime}_{r}:=\{\xi\in C^{*}:\xi\cdot D\varphi(\xi)<r^{2}\}.

Using

\widehat{\pi}=(\operatorname{id},D\phi)_{\#}\bigl(\rho\mathbf{1}_{C}\,\,\mathrm{d}\mathcal{H}^{m}\bigr)=(D\varphi,\operatorname{id})_{\#}\bigl(\rho_{L}^{*}\mathbf{1}_{C^{*}}\,\,\mathrm{d}\xi\bigr),

we obtain

	$\displaystyle(\operatorname{pr}_{1})_{\#}(\widehat{\pi}\lfloor\widehat{G}_{r})$	$\displaystyle=\rho\mathbf{1}_{A_{r}}\,\,\mathrm{d}\mathcal{H}^{m},$
	$\displaystyle(\operatorname{pr}_{2})_{\#}(\widehat{\pi}\lfloor\widehat{G}_{r})$	$\displaystyle=\rho_{L}^{*}\mathbf{1}_{A^{\prime}_{r}}\,\,\mathrm{d}\xi.$

Since $D\phi$ and $D\varphi$ are continuous, the functions

x\longmapsto x\cdot D\phi(x),\qquad\xi\longmapsto\xi\cdot D\varphi(\xi)

are continuous. Hence $A_{r}\subset C$ and $A^{\prime}_{r}\subset C^{*}$ are open. Since $\rho>0$ in $C$ and $\rho_{L}^{*}>0$ in $C^{*}$ ,

\operatorname{spt}\bigl(\rho\mathbf{1}_{A_{r}}\,\,\mathrm{d}\mathcal{H}^{m}\bigr)=\overline{A_{r}},\qquad\operatorname{spt}\bigl(\rho_{L}^{*}\mathbf{1}_{A^{\prime}_{r}}\,\,\mathrm{d}\xi\bigr)=\overline{A^{\prime}_{r}},

where the closures are taken in $L$ . Both supports are compact, being the coordinate projections of $\operatorname{spt}(\widehat{\pi}\lfloor\widehat{G}_{r})$ . Thus $A_{r}$ and $A^{\prime}_{r}$ are bounded.

Moreover, (6.17) and the transport identity give $\int_{A_{r}}\rho\,\,\mathrm{d}\mathcal{H}^{m}=\int_{A^{\prime}_{r}}\rho^{*}_{L}\,\,\mathrm{d}\xi=r^{n}.$ All the hypotheses of Theorem 6.3 are now satisfied. Hence $C,C^{*}$ are cones, $\phi,\varphi$ are two-homogeneous, and $\rho,\rho^{*}_{L}$ are $q$ -homogeneous.

It remains to lift the homogeneity from the projected problem. Denote

\Omega^{*}_{\infty,\xi}:=\Omega_{\infty}^{*}\cap(\xi+L^{\perp}).

Since $\Omega_{\infty}^{*}$ is convex and $0\in\overline{\Omega_{\infty}^{*}}$ ,

t\Omega^{*}_{\infty,\xi}\subset\Omega^{*}_{\infty,t\xi}\qquad(0<t<1).

The $q$ -homogeneity of $\rho^{*}_{L}$ gives $|\Omega^{*}_{\infty,t\xi}|=t^{q}|\Omega^{*}_{\infty,\xi}|.$ The fibers are open convex sets of finite positive volume. Hence, we have

\Omega^{*}_{\infty,t\xi}=t\Omega^{*}_{\infty,\xi}\qquad(0<t<1).

Since $C^{*}$ is a cone, applying this identity at $s\xi$ with $t=s^{-1}$ gives the same equality for $s>1$ . Therefore $\Omega^{*}_{\infty}$ is a cone. Together with (6.16), the fact that $\Omega_{\infty}^{*}$ is a cone shows that

\operatorname{cl}\bigl((\tilde{v}_{\infty}|_{\Omega_{\infty}^{*}})+I_{\Omega_{\infty}^{*}}\bigr)

is two-homogeneous. By (5.2), this function equals $\tilde{v}_{\infty}$ , while $\tilde{u}_{\infty}=\tilde{v}_{\infty}^{*}$ . Hence both $\tilde{v}_{\infty}$ and $\tilde{u}_{\infty}$ are two-homogeneous.

Thus $\tilde{u}_{\infty}$ is two-homogeneous in both alternatives. If $p_{1}$ is the centering vector at height $1$ , then

\sqrt{h}\,S_{1}^{c}[\tilde{u}_{\infty}](0)=\{x:\tilde{u}_{\infty}(x)-\sqrt{h}\,p_{1}\cdot x<h\}.

This set has center of mass zero. By uniqueness of the centering plane,

S^{c}_{h}[\tilde{u}_{\infty}](0)=\sqrt{h}\,S^{c}_{1}[\tilde{u}_{\infty}](0).

Taking $h=1/4$ , and using Lemma 5.1(ii), proves the assertion. ∎

7. Global $W^{2,p}$ estimates

We first record how density oscillations behave under the blow-up normalization. This also shows that one-sided affine normalization preserves the small relative oscillation on the target sub-level sets.

Proposition 7.1.

Suppose $u,v,f,g$ satisfy (1.2) and (1.1). Let $r_{j}\rightarrow 0$ , $x_{j}\in\overline{\Omega}$ , and $y_{j}=Du(x_{j})$ . For each $j$ , apply the normalization of Section 4 at $(x_{j},y_{j})$ with height $h_{j}=r_{j}^{2}$ , choosing the positive definite linear map $A_{j}$ so that

A_{j}(S_{r_{j}^{2}}^{c}[u](x_{j})-x_{j})

is in John position. Denote the resulting potentials, densities, and measures by $\tilde{u}_{j},\tilde{v}_{j},\rho_{j},\rho_{j}^{*},\mu_{j},\nu_{j}$ , and choose the common mass factor so that

\mu_{j}(D_{1}(\tilde{u}_{j},0))=1.

Then, for every fixed $R<\infty$ ,

\operatorname{osc}^{\rm rel}_{D_{R}(\tilde{u}_{j},0)}\rho_{j}+\operatorname{osc}^{\rm rel}_{D_{R}(\tilde{v}_{j},0)}\rho_{j}^{*}\longrightarrow 0.

Moreover, $\mu_{j}$ and $\nu_{j}$ have a common doubling constant depending only on $n$ and $\Lambda/\lambda$ .

Proof.

By the change of variable in Section 4, the multiplicative normalization factors do not affect relative oscillations, and

	$\displaystyle D_{R}(\tilde{u}_{j},0)$	$\displaystyle=A_{j}\bigl(D_{r_{j}R}(u,x_{j})-x_{j}\bigr),$
	$\displaystyle D_{R}(\tilde{v}_{j},0)$	$\displaystyle=r_{j}^{-2}A_{j}^{-t}\bigl(D_{r_{j}R}(v,y_{j})-y_{j}\bigr).$

Let $\omega_{f}$ and $\omega_{g}$ be the moduli of continuity of $f$ and $g$ . By (3.2), for every fixed $R$ and all sufficiently large $j$ ,

	$\displaystyle\operatorname{osc}^{\rm rel}_{D_{R}(\tilde{u}_{j},0)}\rho_{j}$	$\displaystyle\leq\lambda^{-1}\omega_{f}\bigl(C(r_{j}R)^{\sigma}\bigr),$
	$\displaystyle\operatorname{osc}^{\rm rel}_{D_{R}(\tilde{v}_{j},0)}\rho_{j}^{*}$	$\displaystyle\leq\lambda^{-1}\omega_{g}\bigl(C(r_{j}R)^{\sigma}\bigr).$

Both right-hand sides tend to zero as $j\to\infty$ .

Finally, the original measures have a common doubling constant depending only on $n$ and $\Lambda/\lambda$ . Affine changes of variables and multiplication by positive constants preserve the doubling property, which proves the last assertion. ∎

We now return to the original continuous densities. All estimates below are uniform in the base point $x_{0}$ .

Proposition 7.2.

There are $C,c_{0}>0$ such that

(7.1)

F_{x_{0}}(r)\leq C\quad(0<r\leq r_{0}),\qquad F_{x_{0}}(r_{0})\geq c_{0}

for every base point $x_{0}$ . For $0<a<b\leq r_{0}$ , put

\mathfrak{a}_{x_{0}}(a,b)=\log\frac{F_{x_{0}}(a)}{F_{x_{0}}(b)}+C\int_{a}^{b}\frac{\vartheta(s)}{s}\,\,\mathrm{d}s.

This quantity is nonnegative and additive on adjacent intervals. Let $r_{k}=2^{-k}r_{0}$ and

a_{k}=\mathfrak{a}_{x_{0}}(r_{k+1},r_{k}).

Then

(7.2)

\sup_{x_{0}}\sum_{k=0}^{N-1}a_{k}=o(N).

Proof.

By (2.5), (2.6) and (2.8) we have

|D_{r}(u,x_{0})|\,|D_{r}(v,y_{0})|\leq Cr^{2n}.

Since $\lambda\leq f,g\leq\Lambda$ , we have $M(r^{2})^{2}\leq Cr^{2n}$ , proving the first part of (7.1). The second part follows easily from the $C^{1,\alpha_{0}}$ estimate of $u$ in Proposition 2.1.

By (3.3) we have that $\mathfrak{a}_{x_{0}}(a,b)\geq 0$ , and additivity is obvious. A straightforward computation shows

\sum_{k=0}^{N-1}a_{k}=\log\frac{F_{x_{0}}(r_{N})}{F_{x_{0}}(r_{0})}+C\int_{r_{N}}^{r_{0}}\frac{\vartheta(s)}{s}\,\,\mathrm{d}s.

The left side is nonnegative, while (7.1) bounds the logarithm from above uniformly. Since $\vartheta(s)\to 0$ as $s\rightarrow 0$ and $r_{N}=2^{-N}r_{0}$ , splitting the integral over the dyadic intervals $[r_{\ell+1},r_{\ell}]$ gives

\frac{1}{N}\int_{r_{N}}^{r_{0}}\frac{\vartheta(s)}{s}\,\,\mathrm{d}s\longrightarrow 0.

Equivalently,

\int_{r_{N}}^{r_{0}}\frac{\vartheta(s)}{s}\,\,\mathrm{d}s=o(N).

This proves (7.2). ∎

Proposition 7.3.

For every $\varepsilon>0$ there are $C_{\varepsilon},h_{\varepsilon}>0$ , independent of the center, such that

(7.3)

B_{C_{\varepsilon}^{-1}h^{1/2+\varepsilon}}(x_{0})\subset S_{h}^{c}[u](x_{0})\subset B_{C_{\varepsilon}h^{1/2-\varepsilon}}(x_{0}),\qquad 0<h<h_{\varepsilon}.

The same estimate holds for $v$ .

Proof.

Fix $\varepsilon>0$ , and choose $\tau\in(0,1/2)$ , to be specified below. Recall that $r_{k}=2^{-k}r_{0}$ , and put

K_{k}(x_{0}):=S_{r_{k}^{2}}^{c}[u](x_{0})-x_{0}.

For integers $Q\geq 1$ and $k\geq Q$ , set

\Delta_{k,Q}(x_{0}):=\mathfrak{a}_{x_{0}}(r_{k+Q},r_{k-Q}).

We claim that there exist $Q\in\mathbb{N}$ , $\delta>0$ , and $k_{*}\in\mathbb{N}$ , all independent of $x_{0}$ , such that $k\geq\max\{Q,k_{*}\}$ and $\Delta_{k,Q}(x_{0})\leq\delta$ imply

(7.4)

(1-\tau)\tfrac{1}{2}K_{k}(x_{0})\subset K_{k+1}(x_{0})\subset(1+\tau)\tfrac{1}{2}K_{k}(x_{0}).

Suppose not. For $Q=j$ , $\delta=j^{-1}$ , and $k_{*}=2j$ , there exist $x_{j}\in\overline{\Omega}$ and $k_{j}\geq 2j$ such that

(7.5)

\Delta_{k_{j},j}(x_{j})\leq j^{-1},\qquad\eqref{goodKk}\text{ fails at }(x_{j},k_{j}).

Let $(U_{j},V_{j},\mu_{j},\nu_{j})$ be the normalized blow-up of the original problem at $(x_{j},Du(x_{j}))$ and height $r_{k_{j}}^{2}$ , with affine transform $A_{j}$ and common mass factor chosen so that $\mu_{j}(D_{1}(U_{j},0))=1$ . By affine equivariance of centered sections, we have

(7.6)

S_{s}^{c}[U_{j}](0)=A_{j}\bigl(S_{sr_{k_{j}}^{2}}^{c}[u](x_{j})-x_{j}\bigr)\qquad(s>0).

In particular, the sections at heights $1$ and $1/4$ are the images of $K_{k_{j}}(x_{j})$ and $K_{k_{j}+1}(x_{j})$ , respectively.

The normalized mass function satisfies

\widehat{F}_{j}(s):=s^{-n}\mu_{j}(D_{s}(U_{j},0))=\frac{F_{x_{j}}(r_{k_{j}}s)}{F_{x_{j}}(r_{k_{j}})}.

Fix $R>1$ . For all sufficiently large $j$ , we have $R\leq 2^{j}$ . By the nonnegativity and additivity of $\mathfrak{a}_{x_{j}}$ , (7.5) implies, for $R^{-1}\leq s\leq R$ ,

\mathfrak{a}_{x_{j}}\!\left(\min\{r_{k_{j}}s,r_{k_{j}}\},\max\{r_{k_{j}}s,r_{k_{j}}\}\right)\leq j^{-1}.

Consequently,

\sup_{R^{-1}\leq s\leq R}|\log\widehat{F}_{j}(s)|\leq j^{-1}+C(\log R)\sup_{0<t\leq Rr_{k_{j}}}\vartheta(t)\longrightarrow 0,

because $k_{j}\geq 2j$ and hence $r_{k_{j}}\to 0$ . Thus $\widehat{F}_{j}\to 1$ locally uniformly on $(0,\infty)$ , which is (4.6).

Let $\rho_{j},\rho_{j}^{*}$ be the densities of $\mu_{j},\nu_{j}$ . Since $k_{j}\geq 2j$ ,

r_{k_{j}}2^{j}=r_{k_{j}-j}\leq r_{j}\longrightarrow 0.

By the estimates in the proof of Proposition 7.1, we have that

\displaystyle\operatorname{osc}^{\rm rel}_{D_{2^{j}}(U_{j},0)}\rho_{j}+\operatorname{osc}^{\rm rel}_{D_{2^{j}}(V_{j},0)}\rho_{j}^{*}

\displaystyle\leq\lambda^{-1}\omega_{f}\bigl(C(r_{k_{j}}2^{j})^{\sigma}\bigr)+\lambda^{-1}\omega_{g}\bigl(C(r_{k_{j}}2^{j})^{\sigma}\bigr)\longrightarrow 0.

This implies (4.5) at every fixed radius. Moreover, these actual blow-ups satisfy (4.1), (4.2), (4.3), and (4.4), and Proposition 7.1 gives a common doubling constant. Hence, by Theorem 6.4, after passing to a subsequence, we have

\overline{S_{1}^{c}[U_{j}](0)}\longrightarrow K,\qquad\overline{S_{1/4}^{c}[U_{j}](0)}\longrightarrow\tfrac{1}{2}K

in Hausdorff distance. Since $B_{1}\subset S_{1}^{c}[U_{j}](0)\subset B_{n^{3/2}}$ , it follows from the above convergence that, for all large $j$ ,

(1-\tau)\tfrac{1}{2}S_{1}^{c}[U_{j}](0)\subset S_{1/4}^{c}[U_{j}](0)\subset(1+\tau)\tfrac{1}{2}S_{1}^{c}[U_{j}](0).

By (7.6), this is precisely (7.4) at $(x_{j},k_{j})$ , contradicting (7.5). The claim follows.

Fix the resulting $Q,\delta,k_{*}$ . For a fixed center $x_{0}$ , abbreviate $K_{k}=K_{k}(x_{0})$ and $\Delta_{k,Q}=\Delta_{k,Q}(x_{0})$ . By additivity,

\Delta_{k,Q}=\sum_{\ell=k-Q}^{k+Q-1}a_{\ell}.

For $N>2Q$ , define

\mathcal{B}_{N}:=\left\{k\in\{Q,\ldots,N-Q\}:\Delta_{k,Q}>\delta\right\}.

Since each $a_{\ell}$ occurs in at most $2Q$ of the sums $\Delta_{k,Q}$ , Proposition 7.2 gives

\#\mathcal{B}_{N}\leq\frac{2Q}{\delta}\sum_{\ell=0}^{N-1}a_{\ell}=o(N)

uniformly in $x_{0}$ .

Now take

k\in\{\max\{Q,k_{*}\},\ldots,N-Q\}\setminus\mathcal{B}_{N}.

Then $\Delta_{k,Q}\leq\delta$ , and the claim gives (7.4).

At every other scale, Lemma 2.3, applied with $\theta=1/4$ , gives

(7.7)

\frac{1}{4n^{3}}K_{k}\subset K_{k+1}\subset n^{3}K_{k}.

Let $\mathcal{E}_{N}\subset\{0,\ldots,N-1\}$ be the set of indices at which (7.4) has not been obtained. The preceding argument shows that

b_{N}:=\#\mathcal{E}_{N}\leq\#\mathcal{B}_{N}+2Q+k_{*}=o(N)

uniformly in $x_{0}$ .

Iterating (7.4) and (7.7), with $g_{N}=N-b_{N}$ , gives

2^{-g_{N}}(1-\tau)^{g_{N}}(4n^{3})^{-b_{N}}K_{0}\subset K_{N}\subset 2^{-g_{N}}(1+\tau)^{g_{N}}(n^{3})^{b_{N}}K_{0}.

Consequently, for a dimensional constant $C$ ,

(7.8)

2^{-N}e^{-C\tau N-Cb_{N}}K_{0}\subset K_{N}\subset 2^{-N}e^{C\tau N+Cb_{N}}K_{0}.

By Proposition 2.1, $K_{0}$ contains and is contained in balls of uniform radii. Choose $\tau$ sufficiently small and then $N$ sufficiently large so that

C\tau N+Cb_{N}\leq 2\varepsilon(\log 2)N.

Since

r_{N}^{2}=4^{-N}r_{0}^{2},

(7.8) yields

B_{C_{\varepsilon}^{-1}(r_{N}^{2})^{1/2+\varepsilon}}(x_{0})\subset S_{r_{N}^{2}}^{c}[u](x_{0})\subset B_{C_{\varepsilon}(r_{N}^{2})^{1/2-\varepsilon}}(x_{0})

for all sufficiently large $N$ , uniformly in $x_{0}$ .

Finally, if $r_{N+1}^{2}\leq h\leq r_{N}^{2}$ , then $h=\theta r_{N}^{2}$ for some $\theta\in[1/4,1]$ . Lemma 2.3 transfers the preceding estimate from $r_{N}^{2}$ to $h$ , after changing $C_{\varepsilon}$ . This proves (7.3) for $u$ . The proof for $v$ is identical after interchanging the source and target. ∎

Proof of Theorem 1.1.

The global $W^{2,p}$ estimate follows from Proposition 7.3, Caffarelli’s interior $W^{2,p}$ estimate [2], and a covering argument of Savin [24]. ∎

Appendix A Local regularity

The limiting transports arising in Sections 5 and Sections 6 may have infinite total mass. Since only bounded source and target windows are used in the regularity argument, the theorem of Figalli–Jhaveri admits the following locally finite formulation.

A Radon measure $\lambda$ is locally doubling on ellipsoids if, for every ball $B\subset\mathbb{R}^{m}$ , there is a constant $D_{B}\geq 1$ such that

\lambda(E)\leq D_{B}\lambda(\tfrac{1}{2}E)

whenever $E\subset B$ is an ellipsoid with center in $\operatorname{spt}\lambda$ . Here $\tfrac{1}{2}E$ is the concentric half of $E$ . We shall use without further comment the fact that local doubling on ellipsoids implies local doubling on bounded convex sets; see [18, Corollary 2.5].

Proposition A.1.

Let $X,Y\subset\mathbb{R}^{m}$ be open convex sets and let

\mu=a\mathbf{1}_{X}\,\,\mathrm{d}x,\qquad\nu=b\mathbf{1}_{Y}\,\,\mathrm{d}y,\qquad a,b>0\quad\text{a.e.}

be nonzero locally finite Radon measures. Assume that both are locally doubling on ellipsoids. Let $w:X\to\mathbb{R}$ be a finite convex potential function with $(Dw)_{\#}\mu=\nu$ , and let

\underline{w}(x)=\sup_{\begin{subarray}{c}z\in X\\ p\in\partial w(z)\end{subarray}}\{w(z)+p\cdot(x-z)\}

be its lower-semicontinuous minimal extension. Suppose that

\partial\underline{w}(\mathbb{R}^{m})\subset\overline{Y}.

Then $\underline{w}$ is strictly convex and $C^{1}$ in $X$ , and $Dw:X\longrightarrow Y$ is a homeomorphism.

Proof.

The argument is a localization of the proofs of [17, Theorems 1.2 and 1.6]. We point out only the modifications needed for locally finite measures and possibly unbounded supports.

Extend $\mu$ and $\nu$ by zero outside $X$ and $Y$ , respectively. We have $\operatorname{spt}\mu=\overline{X}$ and $\operatorname{spt}\nu=\overline{Y}.$ By the mass-balance formula [17, Lemma 2.1], we know

\mu(E)=\nu\bigl(\partial\underline{w}(E)\bigr)\qquad\text{for every Borel set }E\subset\mathbb{R}^{m}.

Next, we first prove strict convexity. Let $\ell$ be an affine function supporting $w$ at some point $x_{0}\in X$ . After subtracting $\ell$ , we may assume that $w\geq 0$ and $w(x_{0})=0.$ Define $\Sigma:=\{\underline{w}=0\}.$ Note that $\Sigma$ is closed and convex. We prove that it is a singleton by contradiction.

If $\Sigma$ contains a line, then $\partial\underline{w}(\mathbb{R}^{m})$ is contained in a proper affine hyperplane. This contradicts $(D\underline{w})_{\#}\mu=\nu$ , since $b>0$ almost everywhere in the open set $Y$ . Hence $\Sigma$ has an exposed point.

Suppose first that there is an exposed point in $\overline{X}\cap\operatorname{int}(\operatorname{dom}\underline{w}).$ After a translation, we may assume that the exposed point is the origin. Similarly as in [17, Theorem 1.2, Case 2a], choose a bounded open set $U\Subset\operatorname{int}(\operatorname{dom}\underline{w})$ containing the origin and a ball compactly contained in $X$ . Define

\Upsilon:=\operatorname{conv}\overline{\{D\underline{w}(x):x\in U,\ \underline{w}\text{ is differentiable at }x\}}.

Then $\Upsilon$ is compact and convex, and $\partial\underline{w}(U)\subset\Upsilon$ . Define

\Omega_{U}=\partial\underline{w}^{*}(\Upsilon),\qquad\rho=\mu\lfloor\Omega_{U},\qquad\gamma=\nu\lfloor\Upsilon,

and introduce the truncated minimal extension

\phi=\sup_{p\in\Upsilon}\{p\cdot x-\underline{w}^{*}(p)\}.

Then $\phi$ is globally finite and Lipschitz,

\phi=\underline{w}\quad\text{on }\Omega_{U},\qquad\partial\phi(\mathbb{R}^{m})\subset\Upsilon,\qquad(D\phi)_{\#}\rho=\gamma.

Moreover, $0<\rho(\mathbb{R}^{m})=\gamma(\mathbb{R}^{m})<\infty.$ Since $\phi=\underline{w}$ near the origin and $0\leq\phi\leq\underline{w}$ , the origin remains an exposed point of $\{\phi=0\}$ . We may now apply the John-normalized long-cone argument in [17, proof of Theorem 1.2, Case 2a]. All source and target sets occurring there are contained, in the original variables, in fixed bounded sets. Thus only the corresponding local doubling constants of $\mu$ and $\nu$ are used, and the same packing contradiction follows.

Suppose next that the exposed point belongs to $\overline{X}\cap\partial(\operatorname{dom}\underline{w}).$ After the normalization used in [17, proof of Theorem 1.2, Case 2b], let

\underline{w}_{\varepsilon}(x)=\underline{w}(x)-\varepsilon(x_{1}+1),\qquad S_{\varepsilon}=\{\underline{w}_{\varepsilon}\leq 0\},

and let $A_{\varepsilon}$ be the John map of $S_{\varepsilon}$ , with linear part $L_{\varepsilon}$ . If $q$ is an exterior normal to $\operatorname{dom}\underline{w}$ at the exposed point, the same polar construction implies

\operatorname{conv}\bigl(B_{r}\cup(a_{\varepsilon}+\mathbb{R}_{+}\xi_{\varepsilon})\bigr)\subset\partial\widetilde{w}_{\varepsilon}(\widetilde{S}_{\varepsilon}),

where

a_{\varepsilon}=-L_{\varepsilon}^{-t}e_{1},\qquad\xi_{\varepsilon}=\frac{L_{\varepsilon}^{-t}q}{|L_{\varepsilon}^{-t}q|}.

Choose a finite truncated cone whose normalized length tends to infinity, and the images of all target sets used in the packing argument remain in fixed bounded sets in the original target variables. The disjoint long-cone packing argument of [17, (2.3)–(2.7)], using the local source and target doubling constants on those fixed bounded sets, again yields a contradiction.

Finally, suppose that the exposed point does not belong to $\overline{X}$ . We may assume that the exposed point is the origin and that

S_{0}:=\Sigma\cap\{x_{1}\geq-1\}\Subset\mathbb{R}^{m}\setminus\overline{X}.

For

\underline{w}_{\varepsilon}=\underline{w}-\varepsilon(x_{1}+1),\qquad S_{\varepsilon}=\{\underline{w}_{\varepsilon}\leq 0\},

one has $S_{\varepsilon}\Subset\mathbb{R}^{m}\setminus\overline{X}$ for all small $\varepsilon>0$ . The polar estimate from [17, proof of Theorem 1.2, Case 3] implies that

B_{c\varepsilon}\subset\partial\underline{w}_{\varepsilon}(S_{\varepsilon})\subset\overline{Y}-\varepsilon e_{1}.

Consequently, with $\nu_{\varepsilon}=(\operatorname{Id}-\varepsilon e_{1})_{\#}\nu$ ,

0=\mu(S_{\varepsilon})=\nu_{\varepsilon}\bigl(\partial\underline{w}_{\varepsilon}(S_{\varepsilon})\bigr)\geq\nu_{\varepsilon}(B_{c\varepsilon})>0,

a contradiction. Thus $\underline{w}$ is strictly convex in $X$ .

The differentiability argument is the localized version of [17, proof of Theorem 1.6, Part 1]. Fix $x_{0}\in X$ . Suppose that $\partial\underline{w}(x_{0})$ is not a singleton. Since $x_{0}\in\operatorname{int}(\operatorname{dom}\underline{w})$ , $\partial\underline{w}(x_{0})$ is compact. Choosing an exposed point of this compact convex set and applying the affine normalization used in [17, proof of Theorem 1.6, Part 1], $\underline{w}$ is differentiable in $X$ .

It remains to identify the range. Let $v=\underline{w}^{*}|_{Y}$ and let $\underline{v}$ be its minimal extension. Then $v$ is a finite convex function on $Y$ and satisfies

(Dv)_{\#}\nu=\mu,\qquad\partial\underline{v}(\mathbb{R}^{m})\subset\overline{X}.

Applying the preceding argument with source and target interchanged shows that $\underline{v}$ is strictly convex and $C^{1}$ in $Y$ . Thus $T=D\underline{w}|_{X},\ S=D\underline{v}|_{Y}$ are continuous and injective. By invariance of domain, we have

T(X)\subset Y,\qquad S(Y)\subset X.

Moreover, by Legendre duality, we have

S\circ T=\operatorname{id}_{X},\qquad T\circ S=\operatorname{id}_{Y}.

Therefore $T(X)=Y$ , and $D\underline{w}:X\longrightarrow Y$ is a homeomorphism. ∎

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	$\displaystyle\int_{0}^{\infty}\eta(h)A_{C^{*}}(h)\,\,\mathrm{d}h$	$\displaystyle=\lim_{\ell\to\infty}\int_{C^{}}\eta(\Psi^{})\,\rho^{}\,\gamma_{\ell}^{}\,y\cdot\nabla\Psi^{*}\,\,\mathrm{d}y$
		$\displaystyle=\lim_{\ell\to\infty}\int_{C}\eta(\Psi)\,\rho\,\gamma_{\ell}\,\bigl(\Psi+T^{t}\left(D^{2}\phi(x)\right)^{-1}T\bigr)\,\,\mathrm{d}x$
		$\displaystyle=\lim_{\ell\to\infty}\int_{0}^{\infty}\eta(h)\int_{\{\Psi=h\}}\frac{\rho\gamma_{\ell}}{\|\nabla\Psi\|}\bigl(h+T^{t}\left(D^{2}\phi(x)\right)^{-1}T\bigr)\,\,\mathrm{d}\mathcal{H}^{m-1}\,\,\mathrm{d}h$
		$\displaystyle=\int_{0}^{\infty}\eta(h)\int_{\{\Psi=h\}}\frac{\rho}{\|\nabla\Psi\|}\bigl(h+T^{t}\left(D^{2}\phi(x)\right)^{-1}T\bigr)\,\,\mathrm{d}\mathcal{H}^{m-1}\,\,\mathrm{d}h.$

Global W2,pW^{2,p} Regularity in Optimal Transport

Abstract.

1. Introduction

Theorem 1.1.

2. Preliminaries

Proposition 2.1.

Proof.

Lemma 2.2.

Proof.

Lemma 2.3.

Proof.

3. An almost monotonicity estimate

Theorem 3.1.

Proof.

4. Blow-up analysis

Proposition 4.1.

Proof.

Lemma 4.2.

Proof.

Lemma 4.3.

Proof.

5. Limits of normalized transports

Lemma 5.1.

Proof.

Lemma 5.2.

Proof.

Lemma 5.3.

Proof.

Lemma 5.4.

Proof.

6. Rigidity of normalized limits

Lemma 6.1.

Proof.

Lemma 6.2.

Proof.

Theorem 6.3.

Proof.

Theorem 6.4.

Proof.

7. Global W2,pW^{2,p} estimates

Proposition 7.1.

Proof.

Proposition 7.2.

Proof.

Proposition 7.3.

Proof.

Proof of Theorem 1.1.

Appendix A Local regularity

Proposition A.1.

Proof.

References

Global $W^{2,p}$ Regularity in Optimal Transport

7. Global $W^{2,p}$ estimates