Dimension-Free Lipschitz Bounds for Brenier Maps to Compactly Supported Log-Concave Targets

We fix an integer $d\ge 1$ and a symmetric positive-definite matrix $Q\in {\mathbb{R}}^{d\times d}$. Let $V:{\mathbb{R}}^d\to \mathbb{R}$ be finite, set $$Z_{\mu }:={\int }_{{\mathbb{R}}^d}{e}^{-V(x)}dx\in (0,\mathrm{\infty }),d\mu (x):=Z_{\mu }^{-1}{e}^{-V(x)}dx,$$ and assume that $\mu$ has finite second moment and that $$x⟼\frac{1}{2}⟨Qx,x⟩-V(x)$$ is convex. Let $\nu$ be a compactly supported log-concave probability measure with support $K$, and let $\mathrm{\nabla }\mathrm{\Phi }$ be the Brenier map from $\mu$ to $\nu$. For $v\in {\mathbb{R}}^d$, define $$w_K(v):=\underset{y\in K}{\sup }⟨y,v⟩-\underset{y\in K}{\inf }⟨y,v⟩\mathrm{.}$$ We prove that $${\mathrm{\partial }}_{vv}\mathrm{\Phi }\le 0.587\sqrt{⟨Qv,v⟩}{w}_K(v)(v\in {\mathbb{R}}^d)$$ in the sense of distributions. We show that $\mathrm{\nabla }\mathrm{\Phi }$ has an everywhere-defined globally Lipschitz representative such that $$\mathrm{Lip}(\mathrm{\nabla }\mathrm{\Phi })\le 0.587\sqrt{\mathrm{\parallel }Q{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}}\mathrm{diam}(K)\mathrm{.}$$ The directional Hessian estimate is affinely covariant, whereas the global Lipschitz estimate is dimension-free. The result also applies to singular or lower-dimensional targets. In particular, it removes the $\sqrt{d}$ loss in Kolesnikov’s estimate for the Brenier map from Gaussian measure to normalised Lebesgue measure on a convex body. We also prove new bounds that depend only on the support for compactly supported semi-log-concave targets, which includes targets with bounded negative curvature.