On measures of symmetry and floating bodies

TL;DR

The paper introduces a precise measure of symmetry for convex bodies, establishes an upper bound, characterizes equality cases, and relates these findings to convex floating bodies and toric varieties.

Contribution

It provides a new exact upper bound for the symmetry measure of convex bodies and characterizes cases of equality, connecting to recent work on toric varieties.

Findings

Established a precise upper bound on the symmetry measure.

Characterized the equality cases for the symmetry measure.

Connected symmetry measures to convex floating bodies and toric varieties.

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.