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A reverse isoperimetric inequality in three-dimensional space forms

Kostiantyn Drach, Gil Solanes, Kateryna Tatarko

Abstract

A $λ$-convex body in a three-dimensional space form $M^3(c)$ of constant curvature $c$ is a compact convex set $K$ whose boundary $\partial K$ has normal curvatures bounded below by a constant $λ>0$ (in a weak sense). Within this class, we prove a sharp reverse isoperimetric inequality: among all $λ$-convex bodies in $M^3(c)$, with a fixed surface area, the body of minimal volume is the $λ$-convex lens, i.e., the domain bounded by two totally umbilical caps of curvature $λ$. Moreover, this minimizer is unique. This result confirms Borisenko's Conjecture in the three-dimensional model spaces of constant curvature for $c\neq 0$, and complements recent progress on the conjecture in the Euclidean case $c=0$. As a by-product, our method also yields an alternative proof of the corresponding reverse isoperimetric inequality in two-dimensional hyperbolic space.

A reverse isoperimetric inequality in three-dimensional space forms

Abstract

A -convex body in a three-dimensional space form of constant curvature is a compact convex set whose boundary has normal curvatures bounded below by a constant (in a weak sense). Within this class, we prove a sharp reverse isoperimetric inequality: among all -convex bodies in , with a fixed surface area, the body of minimal volume is the -convex lens, i.e., the domain bounded by two totally umbilical caps of curvature . Moreover, this minimizer is unique. This result confirms Borisenko's Conjecture in the three-dimensional model spaces of constant curvature for , and complements recent progress on the conjecture in the Euclidean case . As a by-product, our method also yields an alternative proof of the corresponding reverse isoperimetric inequality in two-dimensional hyperbolic space.
Paper Structure (7 sections, 5 theorems, 47 equations)

This paper contains 7 sections, 5 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $\lambda$-convex bodies
  4. The Gauss--Bonnet theorem in constant curvature 3-space
  5. Inner parallel bodies and variation of the surface area
  6. Proof of the Main Theorem
  7. Proof of the reverse isoperimetric inequality in $\mathbb{H}^2(-1)$

Key Result

Theorem 2.2

Let $K \subset M^n(c)$ be a $\lambda$-convex body. Then for every $p \in \partial K$ there exists a supporting $\lambda$-sphere $\mathcal{S}_{\lambda}(p)$ bounding the $\lambda$-ball $\mathcal{B}_\lambda(p)$ such that $K \subseteq \mathcal{B}_{\lambda}(p)$. ∎

Theorems & Definitions (13)

  • Definition 2.1: $\lambda$-convex body
  • Theorem 2.2: Blaschke's Rolling Theorem, Bla56DrBla
  • Definition 2.3: $\lambda$-convex polyhedron
  • Definition 2.4: $\lambda$-convex lens
  • Theorem 3.1: Gauss--Bonnet for $\lambda$-convex polyhedra
  • proof
  • Theorem 4.1
  • proof
  • Theorem 4.2: Variation of the surface area for inner parallel polyhedra
  • proof
  • ...and 3 more