A reciprocity principle for constrained isoperimetric problems and existence of isoperimetric subregions in convex sets

TL;DR

This paper extends the reciprocity principle for isoperimetric problems to subsets within convex regions and characterizes convex sets in which isoperimetric solutions exist, focusing on relative perimeter in Euclidean space.

Contribution

It proves the reciprocity principle for constrained subsets within convex regions and characterizes convex sets in  ext{D} where isoperimetric solutions exist.

Findings

Reciprocity principle holds for subsets of convex regions in  ext{D}

Characterization of convex sets with isoperimetric solutions

Analysis of relative perimeter in constrained isoperimetric problems

Abstract

It is a well known fact that in $\mathbb{R}^n$ a subset of minimal perimeter $L$ among all sets of a given volume is also a set of maximal volume among all sets of the same perimeter $L$. This is called the reciprocity principle for isoperimetric problems. The aim of this note is to prove this relation in the case where the class of admissible sets is restricted to the subsets of some subregion $G\subsetneq\mathbb{R}^n$. Furthermore, we give a characterization of those (unbounded) convex subsets of $\mathbb{R}^2$ in which the isoperimetric problem has a solution. The perimeter that we consider is the one relative to $\mathbb{R}^n$.