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Some analogues of isoperimetric inequality

Subash Chandra Behera, Shiv Parsad

TL;DR

The paper develops isoperimetric-type inequalities for cyclic and tangential polygons in hyperbolic geometry, establishing sharp bounds that relate perimeter to inradius or circumradius and vice versa, with equality uniquely at regular polygons. A key technical tool is a convexity lemma for minimizing sums of a convex function under a fixed sum constraint, which, together with hyperbolic trigonometry, shows that regular polygons extremize relevant quantities. The authors extend single-polygon results to multiple polygons with constraints on total radius or area, providing explicit formulas for the minimal or maximal sums of perimeters and areas and proving that congruent regular configurations are optimal. The work suggests avenues for further generalization beyond centered or tangential polygons and raises open questions about the robustness of these extremal properties in broader geometries.

Abstract

The discrete isoperimetric inequality states that among all n -gons with a fixed area, the regular n -gon has the least perimeter. We prove analogues of the discrete isoperimetric inequality (involving circumradius or inradius) for cyclic and tangential polygons in hyperbolic geometry, considering both single and multiple polygons. Furthermore, we establish two versions of the isoperimetric inequality for multiple polygons in hyperbolic geometry with some restriction on their area or perimeter.

Some analogues of isoperimetric inequality

TL;DR

The paper develops isoperimetric-type inequalities for cyclic and tangential polygons in hyperbolic geometry, establishing sharp bounds that relate perimeter to inradius or circumradius and vice versa, with equality uniquely at regular polygons. A key technical tool is a convexity lemma for minimizing sums of a convex function under a fixed sum constraint, which, together with hyperbolic trigonometry, shows that regular polygons extremize relevant quantities. The authors extend single-polygon results to multiple polygons with constraints on total radius or area, providing explicit formulas for the minimal or maximal sums of perimeters and areas and proving that congruent regular configurations are optimal. The work suggests avenues for further generalization beyond centered or tangential polygons and raises open questions about the robustness of these extremal properties in broader geometries.

Abstract

The discrete isoperimetric inequality states that among all n -gons with a fixed area, the regular n -gon has the least perimeter. We prove analogues of the discrete isoperimetric inequality (involving circumradius or inradius) for cyclic and tangential polygons in hyperbolic geometry, considering both single and multiple polygons. Furthermore, we establish two versions of the isoperimetric inequality for multiple polygons in hyperbolic geometry with some restriction on their area or perimeter.

Paper Structure

This paper contains 8 sections, 16 theorems, 56 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A convexity lemma
  4. Proof of main results
  5. Isoperimetric type inequalities for a cyclic or tangential polygon
  6. Isoperimetric type inequalties for multiple tangential or cylic polygons
  7. Isoperimetric inequality for multiple polygons
  8. Concluding remarks

Key Result

Theorem 1.1

For any tangential hyperbolic $n$-gon $P$, $\textit{Peri}(P) \geq 2n \tanh^{-1} \left( \tan(\pi/n) \sinh r(P) \right)$, with equality if and only if $P$ is regular.

Figures (3)

  • Figure 1:
  • Figure 2: A triangular section of regular hyperbolic $n$-gon.
  • Figure 3: Tangential and cyclic polygons

Theorems & Definitions (29)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 19 more