A Discrete Four Vertex Theorem for Hyperbolic Polygons

TL;DR

This paper extends the classical four vertex theorem to convex polygons in hyperbolic geometry, providing a discrete analog that broadens understanding of curvature extrema in non-Euclidean settings.

Contribution

It introduces a novel discrete four vertex theorem for convex hyperbolic polygons, adapting existing Euclidean methods to hyperbolic geometry.

Findings

Proves the existence of at least four curvature extrema in convex hyperbolic polygons.

Adapts Musin's Euclidean techniques to hyperbolic geometry.

Establishes a discrete four vertex theorem in a non-Euclidean context.

Abstract

There are many four vertex type theorems appearing in the literature, coming in both smooth and discrete flavors. The most familiar of these is the classical theorem in differential geometry, which states that the curvature function of a simple smooth closed curve in the plane has at least four extreme values. This theorem admits a natural discretization to Euclidean polygons due to O. Musin. In this article we adapt the techniques of Musin and prove a discrete four vertex theorem for convex hyperbolic polygons.