2013/02/08 by Szarek, Stanislaw J.
#46B20 #52A20 #52A40 #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.1302.2076
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.