The Four-Vertex Theorem, The Evolute, and The Decomposition of Polygons

TL;DR

This paper reviews the Four-Vertex Theorem's history, explores its discrete and smooth versions, and introduces new results on polygon decomposition that lead to discrete Four-Vertex Theorems.

Contribution

It introduces the concept of polygon decomposition and demonstrates how it influences extremality, providing new proofs of discrete Four-Vertex Theorems.

Findings

New results on how polygon decomposition affects extremality

Derivation of discrete Four-Vertex Theorems from these results

Overview of the relationship between evolutes and the Four-Vertex Theorem

Abstract

The Four-Vertex Theorem has been of interest ever since a discrete version appeared in 1813 due to Cauchy. Up until now, there have been many different versions of this theorem, both for discrete cases and smooth cases. In 2004, an approach relating the discrete Four-Vertex Theorem to the evolute was published, and here we will give an overview of this paper. We then will define the notion of the decomposition of polygons, and derive some new results about how this notion affects various types of extremality. We will see that from our fresh results we can easily derive discrete Four-Vertex Theorems.