Minimal clusters of four planar regions with the same area

TL;DR

This paper proves that the most perimeter-efficient way to enclose four equal-area regions in a plane involves all regions being connected, with a unique optimal topology.

Contribution

It establishes the minimal perimeter configuration for four equal-area planar regions and proves the uniqueness of its topology.

Findings

Optimal clusters have all regions connected.

The topology of the minimal cluster is uniquely determined.

Minimal perimeter configuration is proven to be optimal.

Abstract

We prove that the optimal way to enclose and separate four planar regions with equal area using the less possible perimeter requires all regions to be connected. Moreover, the topology of such optimal clusters is uniquely determined.