Two properties of optimisers for the reverse isoperimetric problem

TL;DR

This paper investigates the properties of convex sets that maximize perimeter under volume constraints in Riemannian manifolds, revealing curvature bounds and regularity conditions across different geometries.

Contribution

It establishes the non-existence of smooth maximizers with prescribed volume and shows constant principal curvature in certain regions for potential maximizers.

Findings

No $C^{2}$-smooth maximizers exist for the problem.

Maximizers have constant smallest principal curvature where they are $C^{2}$.

Results hold in Euclidean, spherical, and hyperbolic spaces.

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.