Prescription of Gauss curvature using optimal mass transport

TL;DR

This paper presents a novel proof of Alexandrov's theorem on prescribing Gauss curvature for convex sets, utilizing duality and optimal mass transport, avoiding traditional PDE and polyhedral methods.

Contribution

It introduces a new proof technique for Gauss curvature prescription that relies solely on convex set duality and optimal mass transport theory.

Findings

Provides a new proof of Alexandrov's theorem

Avoids PDE and polyhedral methods in the proof

Highlights the role of optimal mass transport in convex geometry

Abstract

In this paper we give a new proof of a theorem by Alexandrov on the Gauss curvature prescription of Euclidean convex sets. This proof is based on the duality theory of convex sets and on optimal mass transport. A noteworthy property of this proof is that it does not rely neither on the theory of convex polyhedra nor on P.D.E. methods (which appeared in all the previous proofs of this result).