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arXiv · 1810.00127

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

Roman Chernov·Kostiantyn Drach·Kateryna Tatarko
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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.

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Roman Chernov, Kostiantyn Drach, Kateryna Tatarko. 2018-09-29. A sausage body is a unique solution for a reverse isoperimetric problem. https://arxiv.org/abs/1810.00127

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