The isoperimetric problem in $2$d domains without necks

TL;DR

This paper characterizes all isoperimetric sets within certain 2D domains without necks, demonstrating convexity properties of the isoperimetric profile and its square, which enhances understanding of geometric optimization in these domains.

Contribution

It provides a complete characterization of isoperimetric sets in 2D domains with no-neck property and establishes convexity properties of the isoperimetric profile and its square.

Findings

Complete characterization of isoperimetric sets in no-neck domains

Convexity of the isoperimetric profile above the largest inscribed ball

Global convexity of the square of the isoperimetric profile

Abstract

We give a complete characterization of all isoperimetric sets contained in a domain of the Euclidean plane, that is bounded by a Jordan curve and satisfies a no-neck property. Further, we prove that the isoperimetric profile of such domain is convex above the volume of the largest ball contained in it, and that its square is globally convex.