  A complete set of N+1 mutually unbiased bases (MUBs) forms a convex polytope
in the N^2-1 dimensional space of NxN Hermitian matrices of unit trace. As a
geometrical object such a polytope exists for all values of N, while it is
unknown whether it can be made to lie within the body of density matrices
unless N=p^k, where p is prime. We investigate the polytope in order to see if
some values of N are geometrically singled out. One such feature is found: It
is possible to select N^2 facets in such a way that their centers form a
regular simplex if and only if there exists an affine plane of order N. Affine
planes of order N are known to exist if N=p^k; perhaps they do not exist
otherwise. However, the link to the existence of MUBs--if any--remains to be
found.
