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

Abstract

We fix an integer $d\ge1$ and a symmetric positive-definite matrix $Q\in\mathbb{R}^{d\times d}$. Let $V:\mathbb{R}^d\to\mathbb{R}$ be finite, set \[ Z_\mu:=\int_{\mathbb{R}^d}e^{-V(x)}\,dx\in(0,\infty), \qquad d\mu(x):=Z_\mu^{-1}e^{-V(x)}\,dx, \] 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 \[ \partial_{vv}\Phi \le 0.587 \sqrt{\langle Qv,v\rangle}\,w_K(v) \qquad(v\in\mathbb{R}^d) \] in the sense of distributions. We show that $\nabla\Phi$ has an everywhere-defined globally Lipschitz representative such that \[ \operatorname{Lip}(\nabla\Phi) \le 0.587 \sqrt{\|Q\|_{\mathrm{op}}}\,\operatorname{diam}(K). \] 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.