  We study a geometric structure of a physical region of neutrino mixing
matrices as part of the unit ball of the spectral norm. Each matrix from the
geometric region is a convex combination of unitary PMNS matrices. The disjoint
subsets corresponding to a different minimal number of additional neutrinos are
described as relative interiors of faces of the unit ball. We determined the
Carath\'eodory's number showing that, at most, four unitary matrices of
dimension three are necessary to represent any matrix from the neutrino
geometric region. For matrices which correspond to scenarios with one and two
additional neutrino states, the Carath\'eodory's number is two and three,
respectively. Further, we discuss the volume associated with different
mathematical structures, particularly with unitary and orthogonal groups, and
the unit ball of the spectral norm. We compare the obtained volumes to the
volume of the region of physically admissible mixing matrices for both the
CP-conserving and CP-violating cases in the present scenario with three
neutrino families and scenarios with the neutrino mixing matrix of dimension
higher than three.
