On supporting hyperplanes to convex bodies

TL;DR

This paper proves a dimensionally quantified existence of supporting hyperplanes near boundary points of convex sets, with applications to regularity in Monge-Ampère equations in optimal transportation.

Contribution

It introduces a new result on supporting hyperplanes with controlled distance relative to the convex set’s thickness, advancing regularity theory in optimal transport.

Findings

Existence of supporting hyperplanes with controlled distance near boundary points

Quantitative relation between hyperplane distance and convex set thickness

Applications to regularity results in Monge-Ampère equations

Abstract

Given a convex set and an interior point close to the boundary, we prove the existence of a supporting hyperplane whose distance to the point is controlled, in a dimensionally quantified way, by the thickness of the convex set in the orthogonal direction. This result has important applications in the regularity theory for Monge-Amp\`ere type equations arising in optimal transportation.