How to recognize convexity of a set from its marginals

TL;DR

This paper establishes that a set of finite perimeter with log-concave marginals on almost every hyperplane must be convex, linking marginal regularity to the set's geometric property.

Contribution

It proves that log-concavity of marginals on almost all hyperplanes implies the convexity of sets with finite perimeter, a novel geometric characterization.

Findings

Log-concave marginals imply convexity for finite perimeter sets

Sets with certain marginal regularities are necessarily convex

Provides a new criterion for convexity based on marginals

Abstract

We investigate the regularity of the marginals onto hyperplanes for sets of finite perimeter. We prove, in particular, that if a set of finite perimeter has log-concave marginals onto a.e. hyperplane then the set is convex.