A sausage body is a unique solution for a reverse isoperimetric problem

TL;DR

This paper solves a reverse isoperimetric problem for a special class of convex bodies called λ-concave bodies, showing that the convex hull of two balls minimizes volume for given surface area, contrasting the classical isoperimetric result.

Contribution

It introduces and solves a reverse isoperimetric problem for λ-concave bodies, establishing the convex hull of two balls as the volume minimizer among such bodies.

Findings

Convex hull of two balls of radius 1/λ minimizes volume for fixed surface area.

Proves a reverse quermassintegral inequality for λ-concave bodies.

Contrasts classical isoperimetric results by identifying a volume minimizer.

Abstract

We consider the class of $\lambda$-concave bodies in $\mathbb R^{n+1}$; that is, convex bodies with the property that each of their boundary points supports a tangent ball of radius $1/\lambda$ that lies locally (around the boundary point) inside the body. In this class we solve a reverse isoperimetric problem: we show that the convex hull of two balls of radius $1/\lambda$ (a sausage body) is a unique volume minimizer among all $\lambda$-concave bodies of given surface area. This is in a surprising contrast to the standard isoperimetric problem for which, as it is well-known, the unique maximizer is a ball. We solve the reverse isoperimetric problem by proving a reverse quermassintegral inequality, the second main result of this paper.