On the structure of the body of states with positive partial transpose

We show that the convex set of separable mixed states of the 2 x 2 system is a body of a constant height. This fact is used to prove that the probability of finding a random state to be separable equals twice the probability-of finding a random boundary state to be separable, provided that the random states are generated uniformly with respect to the Hilbert-Schmidt (Euclidean) measure. An analogous property holds for the set of positive-partial-transpose states for an arbitrary bipartite system.