  Topological and geometrical properties of the set of mixed quantum states in
the N-dimensional Hilbert space are analysed. Assuming that the corresponding
classical dynamics takes place on the sphere we use the vector SU(2) coherent
states and the generalised Husimi distributions to define the Monge distance
between arbitrary two density matrices. The Monge metric has a simple
semiclassical interpretation and induces a non-trivial geometry. Among all pure
states the distance from the maximally mixed state \rho_*, proportional to the
identity matrix, admits the largest value for the coherent states, while the
delocalized 'chaotic' states are close to \rho_*. This contrasts the geometries
induced by the standard (trace, Hilbert-Schmidt or Bures) metrics, where the
distance from \rho_* is the same for all pure states. We discuss possible
physical consequences including unitary time evolution and the process of
decoherence. We introduce also a simplified Monge metric, defined in the space
of pure quantum states, and more suitable for numerical computation.
