Extremal areas of polygons with fixed perimeter

TL;DR

This paper studies the extremal areas of polygons with fixed perimeter, characterizing critical points as regular stars and analyzing their properties within the configuration space.

Contribution

It describes the critical points of the area function on the space of fixed-perimeter polygons and computes their indices when Morse, revealing geometric and topological insights.

Findings

Critical points are regular star polygons.

The area function has the minimal number of critical points.

Indices of critical points are computed when Morse.

Abstract

We consider the configuration space of planar $n$-gons with fixed perimeter, which is diffeomorphic to the complex projective space $\mathbb{C}P^{n-2}$. The oriented area function has the minimal number of critical points on the configuration space. We describe its critical points (these are regular stars) and compute their indices when they are Morse.