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Isoperimetric Inequality for Disconnected Regions

Bidyut Sanki, Arya Vadnere

Abstract

The discrete isoperimetric inequality in Euclidean geometry states that among all $n$-gons having a fixed perimeter $p$, the one with the largest area is the regular $n$-gon. The statement is true in spherical geometry and hyperbolic geometry as well. In this paper, we generalize the discrete isoperimetric inequality to disconnected regions, i.e. we allow the area to be split between regions. We give necessary and sufficient conditions for the result (in Euclidean, spherical and hyperbolic geometry) to hold for multiple $n$-gons whose areas add up.

Isoperimetric Inequality for Disconnected Regions

Abstract

The discrete isoperimetric inequality in Euclidean geometry states that among all -gons having a fixed perimeter , the one with the largest area is the regular -gon. The statement is true in spherical geometry and hyperbolic geometry as well. In this paper, we generalize the discrete isoperimetric inequality to disconnected regions, i.e. we allow the area to be split between regions. We give necessary and sufficient conditions for the result (in Euclidean, spherical and hyperbolic geometry) to hold for multiple -gons whose areas add up.

Paper Structure

This paper contains 8 sections, 11 theorems, 62 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Euclidean and Spherical Geometry
  3. Euclidean Geometry
  4. Spherical Geometry
  5. Hyperbolic Geometry
  6. Technical Lemmas
  7. Proof of Theorem \ref{['thm:Hyp_two']}
  8. Further thoughts

Key Result

Proposition 1.1

Let $P_1, \dots, P_k$ and $P$ be regular $n$-gons (for $n\geq3$) in $\mathbb{M}$, where $\mathbb{M}$ is either $\mathbb{R}^2$ or $\mathbb{S}^2$, with areas $a_1, \dots, a_k$ and $a$ respectively, satisfying $\sum\limits_{i=1}^k a_i = a$. Then

Figures (2)

  • Figure 1: A triangular section of a regular spherical $n$-gon. Here, $A,B$ are two consecutive vertices of the polygon, $O$ is the circumcenter and $M$ is the midpoint of $\overline{AB}$.
  • Figure 2: A triangular section of a regular hyperbolic $n$-gon. Here, $A,B$ are two consecutive vertices of the polygon, $O$ is the circumcenter and $M$ is the midpoint of $\overline{AB}$.

Theorems & Definitions (21)

  • Proposition 1.1
  • Theorem 1.2
  • Theorem 2.1
  • proof
  • Corollary 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • ...and 11 more