Primary Facets Of Order Polytopes

Mixture models on order relations play a central role in recent investigations of transitivity in binary choice data. In such a model, the vectors of choice probabilities are the convex combinations of the characteristic vectors of all order relations of a chosen type. The five prominent types of order relations are linear orders, weak orders, semiorders, interval orders and partial orders. For each of them, the problem of finding a complete, workable characterization of the vectors of probabilities is crucial---but it is reputably inaccessible. Under a geometric reformulation, the problem asks for a linear description of a convex polytope whose vertices are known. As for any convex polytope, a shortest linear description comprises one linear inequality per facet. Getting all of the facet-defining inequalities of any of the five order polytopes seems presently out of reach. Here we search for the facet-defining inequalities which we call primary because their coefficients take only the values -1, 0 or 1. We provide a classification of all primary, facet-defining inequalities of three of the five order polytopes. Moreover, we elaborate on the intricacy of the primary facet-defining inequalities of the linear order and the weak order polytopes

On Compact Hyperbolic Coxeter Polytopes with Few Facets

This thesis is concerned with classifying and bounding the dimension of compact hyperbolic Coxeter polytopes with few facets. We derive a new method for generating the combinatorial type of these polytopes via the classification of point set order types. We use this to complete the classification of d-polytopes with d+4 facets for d=4 and 5. In dimensions 4 and 5, there are 341 and 50 polytopes, respectively, yielding many new examples for further study. By previous work of Felikson and Tumarkin, the only remaining dimension where new polytopes may arise is d=6. We furthermore show that any polytope of dimension 6 must have a missing face of size 3 or 4. The second portion of this thesis provides new upper bounds on the dimension of compact hyperbolic Coxeter d-polytopes with d+k facets for k = 5. In the process of proving the present bounds, we additionally show that there are no compact hyperbolic Coxeter 3-free polytopes of dimension higher than 13. As a consequence of our bounds, we prove that a compact hyperbolic Coxeter 29-polytope has at least 40 facets

Revlex-Initial 0/1-Polytopes

We introduce revlex-initial 0/1-polytopes as the convex hulls of reverse-lexicographically initial subsets of 0/1-vectors. These polytopes are special knapsack-polytopes. It turns out that they have remarkable extremal properties. In particular, we use these polytopes in order to prove that the minimum numbers f(d, n) of facets and the minimum average degree a(d, n) of the graph of a d-dimensional 0/1-polytope with n vertices satisfy f(d, n) <= 3d and a(d, n) <= d + 4. We furthermore show that, despite the sparsity of their graphs, revlex-initial 0/1-polytopes satisfy a conjecture due to Mihail and Vazirani, claiming that the graphs of 0/1-polytopes have edge-expansion at least one.Comment: Accepted for publication in J. Comb. Theory Ser. A; 24 pages; simplified proof of Theorem 1; corrected and improved version of Theorem 4 (the average degree is now bounded by d+4 instead of d+8); several minor corrections suggested by the referee

Revlex-initial 0/1-polytopes

AbstractWe introduce revlex-initial 0/1-polytopes as the convex hulls of reverse-lexicographically initial subsets of 0/1-vectors. These polytopes are special knapsack-polytopes. It turns out that they have remarkable extremal properties. In particular, we use these polytopes in order to prove that the minimum numbers gnfac(d,n) of facets and the minimum average degree gavdeg(d,n) of the graph of a d-dimensional 0/1-polytope with n vertices satisfy gnfac(d,n)⩽3d and gavdeg(d,n)⩽d+4. We furthermore show that, despite the sparsity of their graphs, revlex-initial 0/1-polytopes satisfy a conjecture due to Mihail and Vazirani, claiming that the graphs of 0/1-polytopes have edge-expansion at least one

Compact hyperbolic Coxeter n-polytopes with n+3 facets

We use methods of combinatorics of polytopes together with geometrical and computational ones to obtain the complete list of compact hyperbolic Coxeter n-polytopes with n+3 facets, 3<n<8. Combined with results of Esselmann (1994), Andreev (1970) and Poincare (1882) this gives the classification of all compact hyperbolic Coxeter n-polytopes with n+3 facets.Comment: v4: paper is rewritten. Complete proofs added, errors corrected. 36 pages, a lot of figure