Transformations of measures by Gauss maps

The goal of this paper is to introduce a new class of transformations of measures on R having the form (heuristically) T = φ · ∇φ/|∇φ|, where φ is some function; our construction exhibits an interesting link between two actively developing areas: optimal transportation and geometrical glows (see [1], [2], and [3] concerning these directions). 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 the ball 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, the mapping T is invertible and essentially unique. Our proof employs the optimal transportation techniques. 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. Throughout we assume that d ≥ 2 and denote by H the n-dimensional Hausdorff measure. Let IntA denote the interior of a set A. Given a compact convex set V ⊂ R with boundary M = ∂V , for an arbitrary point x ∈M , let us set