  The totality of normalised density matrices of order N forms a convex set Q_N
in R^(N^2-1). Working with the flat geometry induced by the Hilbert-Schmidt
distance we consider images of orthogonal projections of Q_N onto a two-plane
and show that they are similar to the numerical ranges of matrices of order N.
For a matrix A of a order N one defines its numerical shadow as a probability
distribution supported on its numerical range W(A), induced by the unitarily
invariant Fubini-Study measure on the complex projective manifold CP^(N-1). We
define generalized, mixed-states shadows of A and demonstrate their usefulness
to analyse the structure of the set of quantum states and unitary dynamics
therein.
