On measures of symmetry and floating bodies

Abstract

We consider the following measure of symmetry of a convex n-dimensional body K: $ρ(K)$ 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 $ρ(K)$. It is well known that $ρ(K)=1$ iff K is symmetric. We establish a precise upper bound on $ρ(K)$; 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.

5 pages; this is a slightly edited manuscript from early '00s containing a proof of a 1960 result of Grunbaum; it is being posted since this particular presentation turned out to facilitate certain arguments in toric geometry