# Perimeter approximation of convex discs in the hyperbolic plane and on the sphere

## TL;DR

This paper extends Eggleston's perimeter approximation results from Euclidean geometry to hyperbolic geometry, showing inscribed polygons are optimal in the hyperbolic plane but not on the sphere.

## Contribution

It proves the hyperbolic analogue of Eggleston's theorem and provides a counterexample demonstrating the failure on the sphere.

## Key findings

- Inscribed convex polygons minimize perimeter deviation in hyperbolic plane.
- The same property does not hold on the sphere, as shown by a counterexample.

## Abstract

Eggleston (1957) proved that in the Euclidean plane the best approximating convex $n$-gon to a convex disc $K$ is always inscribed in $K$ if we measure the distance by perimeter deviation. We prove that the analogue of Eggleston's statement holds in the hyperbolic plane, and we give an example showing that it fails on the sphere.