Mass transport generated by a flow of Gauss maps

Let A ⊂ Rd , 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 ˙ x(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. © 2008 Elsevier Inc. All rights reserved.