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Mixed-integer convex representable (MICP-R) sets are those sets that can berepresented exactly through a mixed-integer convex programming formulation.Following up on recent work by Lubin et al. (2017, 2020) we investigatestructural geometric properties of MICP-R sets, which strongly differentiatethem 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 ofconvex sets with countably infinitely many different recession cones. This isin sharp contrast with MILP-R sets which are at most infinite unions ofpolyhedra that share the same recession cone. Second, we provide an example ofan MICP-R set which is the countably infinite union of polytopes all of whichhave different shapes (no pair is combinatorially equivalent, which impliesthey are not affine transformations of each other). Again, this is in sharpcontrast with MILP-R sets which are at most infinite unions of polyhedra thatare all translations of a finite subset of themselves. Interestingly, we showthat a countably infinite union of convex sets sharing the same volume can beMICP-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).

Shapes and recession cones in mixed-integer convex representability