Existence and characterization of regions minimizing perimeter under a volume constraint inside Euclidean cones

TL;DR

This paper investigates the existence and properties of regions with minimal perimeter for a given volume inside Euclidean cones, providing criteria for existence and characterizing stable isoperimetric regions.

Contribution

It establishes conditions for the existence of isoperimetric regions in Euclidean cones and characterizes stable regions as Euclidean balls centered at the cone vertex.

Findings

Nonexistence implies isoperimetric profile matches that of a half-space.

Local convexity ensures existence of isoperimetric regions.

In convex cones, stable regions are Euclidean balls centered at the vertex.

Abstract

We study the problem of existence of regions separating a given amount of volume with the least possible perimeter inside a Euclidean cone. Our main result shows that nonexistence for a given volume implies that the isoperimetric profile of the cone coincides with the one of the half-space. This allows us to give some criteria ensuring existence of isoperimetric regions: for instance, local convexity of the cone at some boundary point. We also characterize which are the stable regions in a convex cone, i.e., second order minima of perimeter under a volume constraint. From this it follows that the isoperimetric regions in a convex cone are the euclidean balls centered at the vertex intersected with the cone.