Isoperimetric Inequality for Disconnected Regions

TL;DR

This paper extends the classical isoperimetric inequality to disconnected regions across Euclidean, spherical, and hyperbolic geometries, providing conditions for multiple polygons with combined areas.

Contribution

It generalizes the isoperimetric inequality to disconnected regions and establishes necessary and sufficient conditions in various geometries.

Findings

Generalization of isoperimetric inequality to disconnected regions

Conditions for multiple polygons with combined areas

Applicability across Euclidean, spherical, and hyperbolic geometries

Abstract

The discrete isoperimetric inequality in Euclidean geometry states that among all $n$-gons having a fixed perimeter $p$, the one with the largest area is the regular $n$-gon. The statement is true in spherical geometry and hyperbolic geometry as well. In this paper, we generalize the discrete isoperimetric inequality to disconnected regions, i.e. we allow the area to be split between regions. We give necessary and sufficient conditions for the result (in Euclidean, spherical and hyperbolic geometry) to hold for multiple $n$-gons whose areas add up.