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

2018/09/29 by Chernov, Roman, Drach, Kostiantyn, Tatarko, Kateryna · 1 citation
#52A30 #52A38 #52A39 #52A40 #52B60 #53A07 #Differential Geometry (math.DG) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1810.00127

Abstract

We consider the class of λ-concave bodies in \mathbb Rⁿ⁺¹; that is, convex bodies with the property that each of their boundary points supports a tangent ball of radius 1/λ 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/λ (a sausage body) is a unique volume minimizer among all λ-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.

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