Embedding $S^n$ into $R^{n+1}$ with given integral Gauss curvature and optimal mass transport on $S^n$

TL;DR

This paper provides a variational solution to Aleksandrov's problem of constructing convex hypersurfaces with prescribed integral Gauss curvature, linking it to optimal mass transport theory on spheres.

Contribution

It introduces a variational approach to solve Aleksandrov's problem and establishes its connection with optimal mass transport on the sphere.

Findings

Existence and uniqueness of convex hypersurfaces with given integral Gauss curvature.

Connection established between Aleksandrov's problem and optimal mass transport theory.

Solution includes the case of convex polytopes.

Abstract

In his book on Convex Polyhedra (section 7.2), A.D. Aleksandrov raised a general question of finding variational statements and proofs of existence of polytopes with given geometric data. The first goal of this paper is to give a variational solution to the problem of existence and uniqueness of a closed convex hypersurface in Euclidean space with prescribed integral Gauss curvature. Our solution includes the case of a convex polytope. This problem was also first considered by Aleksandrov and below it is referred to as Aleksandrov's problem. The second goal of this paper is to show that in variational form the Aleksandrov problem is closely connected with the theory of optimal mass transport on a sphere with cost function and constraints arising naturally from geometric considerations.