Mass transport generated by a flow of Gauss maps

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.