A sausage body is a unique solution for a reverse isoperimetric problem
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 wellknown, 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.
 Publication:

arXiv eprints
 Pub Date:
 September 2018
 arXiv:
 arXiv:1810.00127
 Bibcode:
 2018arXiv181000127C
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Metric Geometry;
 52A30;
 52A38;
 53A07;
 52A39;
 52A40;
 52B60
 EPrint:
 1 figure