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

Abstract

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 \[ Z_μ:=\int_{\mathbb{R}^d}e^{-V(x)}\,dx\in(0,\infty), \qquad dμ(x):=Z_μ^{-1}e^{-V(x)}\,dx, \] and assume that $\mu$ has finite second moment and that \[ x\longmapsto \frac12\langle Qx,x\rangle-V(x) \] is convex. Let $\nu$ be a compactly supported log-concave probability measure with support $K$, and let $\nabla \Phi$ be the Brenier map from $\mu$ to $\nu$. For $v\in {\mathbb{R}}^d$, define \[ w_K(v):= \sup_{y\in K}\langle y,v\rangle - \inf_{y\in K}\langle y,v\rangle. \] We prove that \[ \partial_{vv}Φ\le 0.587 \sqrt{\langle Qv,v\rangle}\,w_K(v) \qquad(v\in\mathbb{R}^d) \] in the sense of distributions. We show that $\nabla \Phi$ has an everywhere-defined globally Lipschitz representative such that \[ \operatorname{Lip}(\nablaΦ) \le 0.587 \sqrt{\|Q\|_{\mathrm{op}}}\,\operatorname{diam}(K). \] 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.