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Two properties of optimisers for the reverse isoperimetric problem

Deniz M. Hamdy, Julian Scheuer

TL;DR

This work studies the reverse isoperimetric problem in space forms $\mathbb{E}^{n+1}$, $\mathbb{S}^{n+1}$, and $\mathbb{H}^{n+1}$. Under the curvature constraint of $\lambda$-convexity, it proves there are no $C^{2}$-maximisers of perimeter for fixed volume, and that on any $C^{2}$-portion of a maximiser the smallest principal curvature is forced to be $\lambda$. The authors combine geometric containment via lens arguments with variational calculus to produce local, volume-preserving area increases unless the curvature bound is saturated. The results unify the Euclidean, spherical, and hyperbolic settings and provide precise curvature-structure information about potential maximisers under $\lambda$-convexity.

Abstract

The reverse isoperimetric problem asks for existence and properties of bounded convex sets in a Riemannian manifold which maximise the perimeter under all those sets of fixed volume which roll freely in a ball of some given radius. If the boundary of the set is of class $C^{2}$, this amounts to a positive lower bound on the principal curvatures and in this class we prove that there are no $C^{2}$-maximisers of perimeter with prescribed volume. In addition, we prove that a given possibly non-$C^{2}$ maximiser has its smallest principal curvature constant in regions where it is of class $C^{2}$. We prove this result in the Euclidean, spherical and hyperbolic space.

Two properties of optimisers for the reverse isoperimetric problem

TL;DR

This work studies the reverse isoperimetric problem in space forms , , and . Under the curvature constraint of -convexity, it proves there are no -maximisers of perimeter for fixed volume, and that on any -portion of a maximiser the smallest principal curvature is forced to be . The authors combine geometric containment via lens arguments with variational calculus to produce local, volume-preserving area increases unless the curvature bound is saturated. The results unify the Euclidean, spherical, and hyperbolic settings and provide precise curvature-structure information about potential maximisers under -convexity.

Abstract

The reverse isoperimetric problem asks for existence and properties of bounded convex sets in a Riemannian manifold which maximise the perimeter under all those sets of fixed volume which roll freely in a ball of some given radius. If the boundary of the set is of class , this amounts to a positive lower bound on the principal curvatures and in this class we prove that there are no -maximisers of perimeter with prescribed volume. In addition, we prove that a given possibly non- maximiser has its smallest principal curvature constant in regions where it is of class . We prove this result in the Euclidean, spherical and hyperbolic space.

Paper Structure

This paper contains 3 sections, 10 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Basic notions and auxiliary results
  3. Proof of the theorem

Key Result

Theorem 1.2

Let $n\geq 1$, $\lambda>0$ and $K$ be a nontrivial compact $\lambda$-convex body in $\Sigma=\mathbb{E}^{n+1}$, $\Sigma=\mathbb{S}^{n+1}$ or $\Sigma=\mathbb{H}^{n+1}$. Suppose that the set $K$ has the largest surface area among all $\lambda$-convex bodies with the same volume as $K$. Then there hold:

Theorems & Definitions (20)

  • Definition 1.1: $\lambda$-convexity
  • Theorem 1.2
  • Definition 2.1: Variation and normal Variation
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • Theorem 2.6
  • ...and 10 more