Mass Transport Generated by a Flow of Gauss Maps

Let A ⊂ R, d ≥ 2, be a compact convex set and let μ = ̺0 dx be a probability measure on A equivalent to the restriction of Lebesgue measure. Let ν = ̺1 dx be a probability measure on Br := {x : |x| ≤ r} equivalent to the restriction of Lebesgue measure. We prove that there exists a mapping T such that ν = μ◦T−1 and T = φ ·n, where φ : A → [0, r] is a continuous potential with convex sub-level sets and n is the Gauss map of the corresponding level sets of φ. Moreover, T is invertible and essentially unique. Our proof employs the optimal transportation techniques. We show that in the case of smooth φ the level sets of φ are governed by the Gauss curvature flow ẋ(s) = −sd−1 ̺1(sn) ̺0(x) K(x) · n(x), where K is the Gauss curvature. As a by-product one can reprove the existence of weak solutions to the classical Gauss curvature flow starting from a convex hypersurface.