A reverse isoperimetric inequality in three-dimensional space forms

TL;DR

This paper establishes a sharp reverse isoperimetric inequality for $ ext{lambda}$-convex bodies in three-dimensional space forms, identifying the minimal volume shape as a $ ext{lambda}$-convex lens and confirming Borisenko's Conjecture for non-zero curvature spaces.

Contribution

The paper proves a new reverse isoperimetric inequality for $ ext{lambda}$-convex bodies in 3D space forms, confirming Borisenko's Conjecture in non-zero curvature spaces and providing an alternative proof in hyperbolic space.

Findings

The minimal volume $ ext{lambda}$-convex body is a $ ext{lambda}$-convex lens.

The result confirms Borisenko's Conjecture for $c eq 0$.

An alternative proof for the hyperbolic case is provided.

Abstract

A $\lambda$-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 $\lambda>0$ (in a weak sense). Within this class, we prove a sharp reverse isoperimetric inequality: among all $\lambda$-convex bodies in $M^3(c)$, with a fixed surface area, the body of minimal volume is the $\lambda$-convex lens, i.e., the domain bounded by two totally umbilical caps of curvature $\lambda$. 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.