A Discrete Four Vertex Theorem for Hyperbolic Polygons

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.