  While faces of a polytope form a well structured lattice, in which faces of
each possible dimension are present, this is not true for general compact
convex sets. We address the question of what dimensional patterns are possible
for the faces of general closed convex sets. We show that for any finite
sequence of positive integers there exist compact convex sets which only have
extreme points and faces with dimensions from this prescribed sequence. We also
discuss another approach to dimensionality, considering the dimension of the
union of all faces of the same dimension. We show that the questions arising
from this approach are highly nontrivial and give examples of convex sets for
which the sets of extreme points have fractal dimension.

Metric Geometry;Optimization and Control