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arXiv · 1302.2076

On measures of symmetry and floating bodies

Stanislaw J. Szarek
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Abstract

We consider the following measure of symmetry of a convex n-dimensional body K: $\rho(K)$ is the smallest constant for which there is a point x in K such that for partitions of K by an n-1-dimensional hyperplane passing through x the ratio of the volumes of the two parts is at most $\rho(K)$. It is well known that $\rho(K)=1$ iff K is symmetric. We establish a precise upper bound on $\rho(K)$; this recovers a 1960 result of Grunbaum. We also provide a characterization of equality cases (relevant to recent results of Nill and Paffenholz about toric varieties) and relate these questions to the concept of convex floating bodies.

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Stanislaw J. Szarek. 2013-02-08. On measures of symmetry and floating bodies. https://arxiv.org/abs/1302.2076

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