Alexandrov's Soap Bubble Theorem for Polygons

TL;DR

This paper characterizes regular polygons as optimal shapes under area constraints for perimeter variations and reviews related recent results and open problems in shape optimization.

Contribution

It extends Alexandrov's Soap Bubble Theorem to polygons, identifying conditions for regular polygons as critical points of perimeter under area constraints.

Findings

Regular polygons are area-constrained critical points of perimeter.

The paper reviews recent shape optimization results for polygons.

It discusses open problems in the field of shape functionals.

Abstract

Regular polygons are characterized as area-constrained critical points of the perimeter functional with respect to particular families of perturbations in the class of polygons with a fixed number of sides. We also review recent results in the literature involving other shape functionals as well as further open problems.