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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 restrictionof 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. Weprove 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 potentialwith convex sub-level sets and ${\rm n}$ is the Gauss map of the correspondinglevel sets of $\phi$. Moreover, $T$ is invertible and essentially unique. Ourproof employs the optimal transportation techniques. We show that in the caseof smooth $\phi$ the level sets of $\phi$ are driven by the Gauss curvatureflow $\dot{x}(s) = -s^{d-1} \frac{\varrho_1(s {\rm n})}{\varrho_0(x)} K(x)\cdot {\rm n}(x)$, where $K$ is the Gauss curvature. As a by-product one canreprove the existence of weak solutions of the classical Gauss curvature flowstarting from a convex hypersurface.

Mass transport generated by a flow of Gauss maps