Hyperbolic polygons of minimal perimeter with given angles

TL;DR

This paper proves that among convex hyperbolic polygons with fixed angles, the one with an inscribed circle has the minimal perimeter, establishing a unique minimal configuration.

Contribution

It demonstrates the minimal perimeter property for convex hyperbolic polygons with given angles, extending geometric optimization results in hyperbolic geometry.

Findings

The inscribed circle polygon uniquely minimizes perimeter.

The proof builds on Schlenker's work in hyperbolic geometry.

The result characterizes optimal polygons with fixed angles.

Abstract

We prove that, among all convex hyperbolic polygons with given angles, the perimeter is minimized by the unique polygon with an inscribed circle. The proof relies on work of J.-M.\ Schlenker.