Mass transport generated by a flow of Gauss maps

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 $B_r:=\{x:|x|\le r\}$ equivalent to the restriction of Lebesgue measure. We prove that there exists a mapping $T$ such that $\nu =\mu \circ T^{-1}$ and $T=\varphi \cdot \mathrm{n}$, where $\varphi :A\to [0,r]$ is a continuous potential with convex sub-level sets and $\mathrm{n}$ is the Gauss map of the corresponding level sets of $\varphi$. Moreover, $T$ is invertible and essentially unique. Our proof employs the optimal transportation techniques. We show that in the case of smooth $\varphi$ the level sets of $\varphi$ are driven by the Gauss curvature flow $\dot{x}(s)=-s^{d-1}\frac{{\varrho }_1(s\mathrm{n})}{{\varrho }_0(x)}K(x)\cdot \mathrm{n}(x)$, 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.

15 pages; minor changes