Shapes and recession cones in mixed-integer convex representability

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. (in: Eisenbrand (ed) Integer Programming and Combinatorial Optimization - 19th International Conference, Springer, Waterloo), (Math. Oper. Res. 47:720-749, 2022) we investigate structural geometric properties of MICP-R sets, which strongly differentiate them from the class of mixed-integer linear representable (MILP-R) sets. 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 (countable) 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 (countable) unions of polyhedra that are all translations of a finite subset of themselves.