We provide a numerical refutation of the developments of Fiorini et al. (2015) * for modelswith disjoint sets of descriptive variables. We also provide an insight into the meaning of the existence of a one-to-one linear map between solutions of such models.

In this paper we extend recent results of Fiorini et al. on the extension complexity of the cut polytope and related polyhedra. We first describe a lifting argument to show exponential extension complexity for a number of NP-complete problems including subset-sum and three dimensional matching. We then obtain a relationship between the extension complexity of the cut polytope of a graph and that of its graph minors. Using this we are able to show exponential extension complexity for the cut polytope of a large number of graphs, including those used in quantum information and suspensions of cubic planar graphs.

The extension complexity of a polytope P is the smallest integer k such that P is the projection of a polytope Q with k facets. We study the extension complexity of n-gons in the plane. First, we give a new proof that the extension complexity of regular n-gons is O(log n), a result originating from work by Ben-Tal and Nemirovski (2001). Our proof easily generalizes to other permutahedra and simplifies proofs of recent results by Goemans (2009), andKaibel and. Second, we prove a lower bound of √ 2n on the extension complexity of generic n-gons. Finally, we prove that there exist n-gons whose vertices lie on a O(n) × O(n 2 ) integer grid with extension complexity Ω( √ n/ √ log n).

For polytopes P 1 , P 2 ⊂ R d we consider the intersection P 1 ∩ P 2 , the convex hull of the union CH(P1 ∪ P2), and the Minkowski sum P1 + P2.For Minkowski sum we prove that enumerating the facets of P1 +P2 is NPhard if P 1 and P 2 are specified by facets, or if P 1 is specified by vertices and P2 is a polyhedral cone specified by facets. For intersection we prove that computing the facets or the vertices of the intersection of two polytopes is NP-hard if one of them is given by vertices and the other by facets.Also, computing the vertices of the intersection of two polytopes given by vertices is shown to be NP-hard. Analogous results for computing the convex hull of the union of two polytopes follow from polar duality. All of the hardness results are established by showing that the appropriate decision version, for each of these problems, is NP-complete .