  Mixed-integer convex representable (MICP-R) sets are those sets that can be
represented exactly through a mixed-integer convex programming formulation.
Following up on recent work by Lubin et al. (2017, 2020) we investigate
structural geometric properties of MICP-R sets, which strongly differentiate
them from the class of mixed-integer linear representable sets (MILP-R). First,
we provide an example of an MICP-R set which is the countably infinite union of
convex sets with countably infinitely many different recession cones. This is
in sharp contrast with MILP-R sets which are at most infinite unions of
polyhedra that share the same recession cone. Second, we provide an example of
an MICP-R set which is the countably infinite union of polytopes all of which
have different shapes (no pair is combinatorially equivalent, which implies
they are not affine transformations of each other). Again, this is in sharp
contrast with MILP-R sets which are at most infinite unions of polyhedra that
are all translations of a finite subset of themselves. Interestingly, we show
that a countably infinite union of convex sets sharing the same volume can be
MICP-R only if the sets are all translations of a finite subset of themselves
(i.e. the natural conceptual analogue to the MILP-R case).
