Isoperimetric inequalities in Euclidean convex bodies

TL;DR

This paper investigates isoperimetric inequalities within convex bodies, establishing convergence properties, profile continuity, and boundary behavior for small volumes without boundary regularity assumptions.

Contribution

It proves the equivalence of Hausdorff and Lipschitz convergence, and analyzes the isoperimetric profile and regions for small volumes in convex bodies.

Findings

Hausdorff and Lipschitz convergence are equivalent.

Isoperimetric profile is continuous with respect to Hausdorff distance.

Behavior of isoperimetric regions for small volume is characterized.

Abstract

In this paper we consider the problem of minimizing the relative perimeter under a volume constraint in the interior of a convex body, i.e., a compact convex set in Euclidean space with interior points. We shall not impose any regularity assumption on the boundary of the convex set. Amongst other results, we shall prove the equivalence between Hausdorff and Lipschitz convergence, the continuity of the isoperimetric profile with respect to the Hausdorff distance,and the convergence in Hausdorff distance of sequences of isoperimetric regions and their free boundaries. We shall also describe the behavior of the isoperimetric profile for small volume, and the behavior of isoperimetric regions for small volume.