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Extremal areas of polygons with fixed perimeter

Giorgi Khimshiashvili, Gaiane Panina, Dirk Siersma

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.

Extremal areas of polygons with fixed perimeter

Abstract

We consider the configuration space of planar -gons with fixed perimeter, which is diffeomorphic to the complex projective space . 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.

Paper Structure

This paper contains 3 sections, 3 theorems, 14 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem 1
  3. Concluding remarks

Key Result

Theorem 1

Figures (2)

  • Figure 1: Regular stars for $n=7$ with their winding numbers.
  • Figure 2: Regular stars for $n=8$ with positive winding numbers.

Theorems & Definitions (3)

  • Theorem 1
  • Lemma 1
  • Lemma 2