  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.
