Stratificational and antipodean properties of boundary states for N x N density matrices

We investigate the space of N x N dimensional density matrices. We show that there exist strata such that boundary states \rho_{p} with p zero eigenvalues lie on or outside the spheres with radii r_{p}=\sqrt{p/N(N-p)}. Moreover, we show that if in a certain direction there is a boundary state with q=N-p equal eigenvalues, then in the opposite (antipodean) direction exists a boundary state with p=N-q equal eigenvalues.