Rigidity of equality cases in Steiner's perimeter inequality

TL;DR

This paper characterizes when equality cases in Steiner's perimeter inequality are rigid, meaning all are vertical translations of the symmetral, using measure-theoretic connectedness and barycenter analysis.

Contribution

It introduces a measure-theoretic notion of connectedness and analyzes barycenter functions to characterize rigidity in Steiner's perimeter equality cases.

Findings

Provides a characterization of equality cases in Steiner's perimeter inequality.

Establishes conditions for rigidity where all equality cases are vertical translations.

Uses measure-theoretic tools to analyze sets of finite perimeter with segmental sections.

Abstract

Characterization results for equality cases and for rigidity of equality cases in Steiner's perimeter inequality are presented. (By rigidity, we mean the situation when all equality cases are vertical translations of the Steiner's symmetral under consideration.) We achieve this through the introduction of a suitable measure-theoretic notion of connectedness and a fine analysis of barycenter functions for sets of finite perimeter having segments as orthogonal sections with respect to an hyperplane.