This is an announcement for the paper "On measures of symmetry and floating bodies" by Stanislaw J. Szarek.

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.

Archive classification: math.MG math.FA

Mathematics Subject Classification: 52A20, 52A40, 46B20

Remarks: 5 pages; this is a slightly edited manuscript from early '00s

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1302.2076

or

http://arXiv.org/abs/1302.2076