Mass transport generated by a flow of Gauss maps

Let $A subset mathbb{R}^d$, $dge 2$, 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 $ u = varrho_1 dx$ be a probability measure on $B_r := {xcolon |x| le r}$ equivalent to the restriction of Lebesgue measure. We prove that there exists a mapping $T$ such that $ u = mu circ T^{-1}$ and $T = phi cdot {rm n}$, where $phicolon A to [0,r]$ 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.