On the local geometry of maps with c-convex potentials

Abstract

We identify a condition for regularity of optimal transport maps that requires only three derivatives of the cost function, for measures given by densities that are only bounded above and below. This new condition is equivalent to the weak Ma–Trudinger–Wang condition when the cost is $C^4$. Moreover, we only require (non-strict) $c$-convexity of the support of the target measure, removing the hypothesis of strong $c$-convexity in a previous result of Figalli et al., but at the added cost of assuming compact containment of the supports of both the source and target measures.

Appendix: Inward pointing normals of convex sets

The results in this appendix are necessary to obtain the lower bound (5.9) on the family of line segments in Lemma 5.5. In turn, this bound is needed to apply the Aleksandrov estimate Theorem 4.1 in the proofs of Theorem 5.7 and Lemma 5.9.

The idea is the following. We have freedom in choosing the direction $v_{0}$ to apply Lemma 5.5. However, in order to obtain a strictly positive lower bound (5.8), we must be careful to select a $v_{0}$ for which the negative actually points into the sublevel set $S_0$. Since it is not a priori obvious that such a choice of direction exists, this is what we aim to show. We note here that the main result of [12] by Figalli et al. plays an analogous role in their paper [11].

We begin by stating a well-known result in convex analysis, the Fenchel–Rockafellar Duality Theorem. Throughout the section, we will fix an $n$-dimensional inner product space $V$ with an inner product $(\cdot , \cdot )$.

Theorem 7.1

(Fenchel–Rockafellar Duality Theorem [29]) If $f$ and $g$ are convex functions on $V$ such that one of the functions is continuous at some point in $\{p\in V\mid f(p)+g(p)<+\infty \}$, then

$$\begin{aligned} \inf _{p\in V}{(f(p)+g(p))}=\max _{p^*\in V}{(-f^*(-p^*)-g^*(p^*))}, \end{aligned}$$

where $f^*$ is the usual Legendre–Fenchel transform,

$$\begin{aligned} f^*(p^*):=\sup _{p\in V}{[(p^*,p)-f(p) ]}. \end{aligned}$$

We will also need the concept of the indicator function of a set.

Definition 7.2

If $G$ is a set, the indicator function of $G$ is defined by

$$\begin{aligned} \delta _G(p):= \left\{ \begin{array}{l@{\quad }l} 0,&{}p\in G\\ +\infty ,&{}p\not \in G. \end{array}\right. \end{aligned}$$

If $G$ is convex and nonempty, $\delta _G$ is a proper, convex function, and if $G$ is closed $\delta _G$ is lower semi-continuous.

We first show essentially the desired result, but applied to the strict normal cone of a convex set with nonempty interior. The added structure of the cone allows us to obtain the result more easily.

Lemma 7.3

Suppose that $\mathcal {A}$ is a convex subset of an $V$ with nonempty interior, and that $p_e\in \partial \mathcal {A}$ is an exposed point of $\mathcal {A}$ (recall Definition 5.1). Then, there exists some §$w^{}_{0}\in N_{p_e}(\mathcal {A})\cap ^{n-1}$ such that $( w^{}_{0}, p) >0$ for all §$p\in N_{p_e}(\mathcal {A})\cap ^{n-1}$. Here §$^{n-1}$ is the unit sphere in $V$.

Proof

We may make a translation to assume that $p_e=0$. First we will show that $N_0(\mathcal {A})$ can be generated by its intersection with some plane that does not intersect the origin. Since $\mathcal {A}^{{{\mathrm{int}}}}\ne \emptyset$, we may assume that for some radius $r_0>0$ and center $p_0\ne 0$, there exists a ball $B_{r_0}(p_0)\subset \mathcal {A}{\setminus }\{0\}$. Now let $K$ be the cone generated by this ball, with $0$ as the vertex, i.e.

$$\begin{aligned} K:=\{\lambda p\mid \lambda \ge 0,\ p\in B_{r_0}(p_0)\}. \end{aligned}$$

Since clearly $K\subset \mathcal {A}$ and $0\in \partial K$, we immediately see that $N_0(\mathcal {A})\subset N_0(K)$, and the normal cone $N_0(K)$ is a cone with vertex $0$ and axial direction $v_{0}:=-\frac{p_0}{\left|p_0\right|}$. Any $(n-1)$-dimensional hyperplane through $0$ that has normal vector orthogonal to $v_{0}$ would contain the point $p_0$, hence cannot be supporting to $K$. In other words, $( v_{0}, w^{}_{})>0$ for any $w^{}_{}\in N_0(K){\setminus } \{0\}$, and in particular, for any $w^{}_{}\in N_0(\mathcal {A}){\setminus }\{0\}$. Thus by the homogeneity of normal cones, we see that the intersection $G:=N_0(\mathcal {A})\cap \{p\in V\mid (p, v_{0})= 1\}$ generates $N_0(\mathcal {A})$. Moreover, it is easy to see that $G$ is compact, convex, and does not contain $0$.

We will now obtain the desired $w^{}_{0}$. Define the concave function $h(p^*):=\inf _{p\in G}{(p^*, p)}$. We wish to choose $f$ and $g$ in Theorem 7.1 so that $g^*$ is the indicator function of $G$, while $-f^*(-p^*)=h(p^*)$. To this end, define

$$\begin{aligned} f(p)&:= \delta _G(p),\\ g(p)&:= \sup _{p^*\in G}{(p^*, p)}=(\delta _G)^*(p). \end{aligned}$$

Since $G$ is convex and nonempty, both $f$ and $g$ are proper convex functions. Moreover, since $G$ is compact we see that $g$ is continuous and finite everywhere, thus we may apply Theorem 7.1 to $f$ and $g$. We can also calculate that indeed, $-f^*(-p^*)=h(p^*)$, while

$$\begin{aligned} g^*(p^*)=(\delta _G)^{**}(p^*)=\delta _G(p^*), \end{aligned}$$

since $\delta _G$ is lower semi-continuous by the closedness of $G$. Hence, we find

$$\begin{aligned} \max _{p^*\in G}{h(p^*)}&= \max _{p^*\in V}{(-f^*(-p^*)-g^*(p^*))}\\&= \inf _{p\in V}{(f(p)+g(p))}\\&= \inf _{p\in G}{\sup _{p^*\in G}{(p^*, p)}}\ge c>0, \end{aligned}$$

since $d(0, G)>0$. By letting $w^{}_{0}$ be the vector achieving the maximum in the expression on the left, normalized to unit length, we obtain the claimed properties. $\square$

Finally, we can use a separation theorem to translate the above lemma into our main result.

Lemma 7.4

Suppose that $\mathcal {A}\subset V$ is convex and contains more than one point, and $p_e\in \partial \mathcal {A}$ is an exposed point of $\mathcal {A}$. Then, there exists some §$v_{0}\in N^0_{p_e}(\mathcal {A})\cap ^{n-1}$ and $\lambda _0>0$ such that $p_e-\lambda v_{0}\in \mathcal {A}$ for any $\lambda \in (0, \lambda _0]$.

Proof

Again, assume that $p_e=0$. Since $\mathcal {A}$ contains more than one point, its affine dimension must be strictly bigger than $0$. If the affine dimension of $\mathcal {A}$ is strictly less than $n$, we may consider the orthogonal projection of $\mathcal {A}$ onto its affine hull for the following proof, so without loss of generality assume that $\mathcal {A}$ has affine dimension $n$. In particular, $\mathcal {A}^{{{\mathrm{int}}}}\ne \emptyset$ and we may choose an associated §$w^{}_{0}\in N_{0}(\mathcal {A})\cap ^{n-1}$ with the property described in Lemma 7.3 above. Our claim will be proven if we can show that $-N^0_{0}(\mathcal {A})\cap \mathcal {A}^{{{\mathrm{int}}}}\ne \emptyset$. Suppose this does not hold. Then by applying the separation theorems [30, Theorem 11.3 and 11.7] to $-N^0_{0}(\mathcal {A})$ and $\mathcal {A}$, we obtain a unit length §$v_{0}\in ^{n-1}$ such that

$$\begin{aligned} (v_{0}, w^{}_{})&\ge 0,\quad \forall w^{}_{}\in -N^0_{0}(\mathcal {A}),\\ (p,v_{0})&\le 0,\quad \forall p\in \mathcal {A}. \end{aligned}$$

Since $0$ is an exposed point of $\mathcal {A}$ we have $N^0_{0}(\mathcal {A})\ne \emptyset$, and hence it can be seen that $(N^0_{0}(\mathcal {A}))^{{{\mathrm{cl}}}}=N_{0}(\mathcal {A})$. Thus as a result of the first inequality above, $( v_{0}, w^{}_{0})\le 0$. However, the second inequality implies that $v_{0}\in N_0(\mathcal {A})$, which contradicts the choice of $w^{}_{0}$, and we obtain the lemma. $\square$