Mass transport generated by a flow of Gauss maps

TL;DR

This paper constructs a mass transport map using Gauss maps for probability measures on convex sets, linking optimal transport with curvature flows and providing new insights into the structure of such mappings.

Contribution

It introduces a novel mass transport map involving Gauss maps and convex potentials, connecting optimal transport with Gauss curvature flow techniques.

Findings

Existence of a transport map with Gauss map structure

Invertibility and essential uniqueness of the map

Connection between level set evolution and Gauss curvature flow

Abstract

Let $A \subset \mathbb{R}^d$, $d\ge 2$, be a compact convex set and let $\mu = \varrho_0 dx$ be a probability measure on $A$ equivalent to the restriction of Lebesgue measure. Let $\nu = \varrho_1 dx$ be a probability measure on $B_r := \{x\colon |x| \le r\}$ equivalent to the restriction of Lebesgue measure. We prove that there exists a mapping $T$ such that $\nu = \mu \circ T^{-1}$ and $T = \phi \cdot {\rm n}$, where $\phi\colon A \to [0,r]$ is a continuous potential with convex sub-level sets and ${\rm n}$ is the Gauss map of the corresponding level sets of $\phi$. Moreover, $T$ is invertible and essentially unique. Our proof employs the optimal transportation techniques. We show that in the case of smooth $\phi$ the level sets of $\phi$ are driven by the Gauss curvature flow $\dot{x}(s) = -s^{d-1} \frac{\varrho_1(s {\rm n})}{\varrho_0(x)} K(x) \cdot {\rm n}(x)$, where $K$ is the Gauss curvature. As a by-product one can reprove the existence of weak solutions of the classical Gauss curvature flow starting from a convex hypersurface.