Shapes and recession cones in mixed-integer convex representability

Mixed-integer convex representable (MICP-R) sets are those that can be represented exactly through a mixed-integer 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 linear (MILP-R) sets. First, provide an example set is countably infinite union with infinitely many different recession cones. This in sharp contrast MILP-R (countable) unions polyhedra share same cone. Second, polytopes all have shapes (no pair combinatorially equivalent, implies they not affine transformations each other). Again, this translations finite subset themselves.