We prove that there are 0/1 polytopes P ⊆ R that do not admit a compact LP formulation. More precisely we show that for every n there is a sets X ⊆ {0, 1}n such that conv(X) must have extension complexity at least 2. In other words, every polyhedron Q that can be linearly projected on conv(X) must have exponentially many facets. In fact, the same result also applies if conv(X) is restricted to ...

We discuss the problem of computing points of IR whose convex hull contains the Euclidean ball, and is contained in a small multiple of it. Given a polytope containing the Euclidean ball, we introduce its successor obtained by intersection with all tangent spaces to the Euclidean ball, whose normals point towards the vertices of the polytope. Starting from the L∞ ball, we discuss the computatio...

An abstract polytope of rank n is said to be chiral if its automorphism group has two orbits on the flags, such that adjacent flags belong to distinct orbits. Examples of chiral polytopes have been difficult to find. A “mixing” construction lets us combine polytopes to build new regular and chiral polytopes. By using the chirality group of a polytope, we are able to give simple criteria for whe...

Let X ↪→ P be a smooth, linearly normal algebraic variety. It is shown that the Mabuchi energy of (X,ωFS |X) restricted to the Bergman metrics is completely determined by the X-hyperdiscriminant of format (n− 1) and the Chow form of X . As a corollary it is shown that the Mabuchi energy is bounded from below for all degenerations in G if and only if the hyperdiscriminant polytope dominates the ...

We reconstruct an n-dimensional convex polytope from the knowledge of its directional moments up to a certain order. The directional moments are related to the projection of the polytope vertices on a particular direction. To extract the vertex coordinates from the moment information we combine established numerical algorithms such as generalized eigenvalue computation and linear interval inter...

We present new facets for the linear ordering polytope. These new facets generalize facets induced by subgraphs called fences, introduced by Grijtschel et al. (1985) and augmented fences, introduced by McLennan (1990). One novelty of the facets introduced here is that each subgraph induces not one but a family of facets, which are not generally rank inequalities. Indeed, we provide the smallest...

Gomory’s and Chvátal’s cutting-plane procedure proves recursively the validity of linear inequalities for the integer hull of a given polyhedron. The number of rounds needed to obtain all valid inequalities is known as the Chvátal rank of the polyhedron. It is well-known that the Chvátal rank can be arbitrarily large, even if the polyhedron is bounded, if it is of dimension 2, and if its intege...

The extension complexity xc(P ) of a polytope P is the minimum number of facets of a polytope that affinely projects to P . Let G be a bipartite graph with n vertices, m edges, and no isolated vertices. Let STAB(G) be the convex hull of the stable sets of G. Since G is perfect, it is easy to see that n 6 xc(STAB(G)) 6 n+m. We improve both of these bounds. For the upper bound, we show that xc(ST...

We study properties of the volume projections n-dimensional cross-polytope ?n={x?Rn?|x1|+?+|xn|?1}. prove that projection ?n onto a k-dimensional coordinate subspace has maximum possible for k=2 and k=3. obtain exact lower bound on such two-dimensional plane. Also, we show there exist local maxima which are not global ones any n>k?2.

A polytopal digraph G(P ) is an orientation of the skeleton of a convex polytope P . The possible non-degenerate pivot operations of the simplex method in solving a linear program over P can be represented as a special polytopal digraph known as an LP digraph. Presently there is no general characterization of which polytopal digraphs are LP digraphs, although four necessary properties are known...