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 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- September 2018
- DOI:
- 10.48550/arXiv.1810.00127
- arXiv:
- arXiv:1810.00127
- Bibcode:
- 2018arXiv181000127C
- Keywords:
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- Mathematics - Differential Geometry;
- Mathematics - Metric Geometry;
- 52A30;
- 52A38;
- 53A07;
- 52A39;
- 52A40;
- 52B60
- E-Print:
- 1 figure