The standard lens cluster in R^2 uniquely minimizes relative perimeter

TL;DR

This paper proves that the standard lens-shaped cluster uniquely minimizes the perimeter in a specific infinite-area partitioning problem in the plane, establishing its optimality and local minimality.

Contribution

It demonstrates the uniqueness and local minimality of the standard lens cluster for the isoperimetric problem with mixed finite and infinite area constraints in the plane.

Findings

The lens cluster is the only local perimeter minimizer under the given constraints.

The lens cluster configuration involves circular arcs meeting at 120 degrees.

The result extends understanding of isoperimetric problems with infinite area regions.

Abstract

In this article we consider the isoperimetric problem for partitioning the plane into three disjoint domains, one having unit area and the remaining two having infinite area. We show that the only solution, up to rigid motions of the plane, is a lens cluster consisting of circular arcs containing the finite area region, attached to a single axis, with two triple junctions where the arcs meet at 120 degree angles. In particular, we show that such a configuration is a local minimizer of the total perimeter functional, and on the other hand any local minimizer of perimeter among clusters with the given area constraints must coincide with a lens cluster having this geometry. Some known results and conjectures on similar problems with both finite and infinite area constraints are presented at the conclusion.