  The set of quantum states consists of density matrices of order $N$, which
are hermitian, positive and normalized by the trace condition. We analyze the
structure of this set in the framework of the Euclidean geometry naturally
arising in the space of hermitian matrices. For N=2 this set is the Bloch ball,
embedded in $\mathbb R^3$. For $N \geq 3$ this set of dimensionality $N^2-1$
has a much richer structure. We study its properties and at first advocate an
apophatic approach, which concentrates on characteristics not possessed by this
set. We also apply more constructive techniques and analyze two dimensional
cross-sections and projections of the set of quantum states. They are dual to
each other. At the end we make some remarks on certain dimension dependent
properties.
