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 $\mu = \varrho_0 dx$beaprobabilitymeasureon$A$ 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 $\mathrm{n}$ 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 $\dot{x}(s) = -s^{d-1} \frac{\varrho_1(s {\rm n})}{\varrho_0(x)} K(x) \cdot {\rm n}(x)$,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.