The g-theorem matrices are totally nonnegative

The g-theorem proved by Billera, Lee, and Stanley states that a sequence is the g-vector of a simplicial polytope if and only if it is an M-sequence. For any d-dimensional simplicial polytope the face vector is gMd where Md is a certain matrix whose entries are sums of binomial coefficients. Björner found refined lower and upper bound theorems by showing that the (2×2)-minors of Md are nonnegative. He conjectured that all minors of Md are nonnegative and that is the result of this note.     

Permanental Polytopes of Doubly Stochastic Matrices

A subpolytope Γ of the polytope Ω_n of all n×n nonnegative doubly stochastic matrices is said to be a permanental polytope if the permanent function is constant on Γ. Geometrical properties of permanental polytopes are investigated. No matrix of the form 1⊕A where A is in Ω₂ is contained in any permanental polytope of Ω₃ with positive dimension. There is no 3-dimensional permanental polytope of Ω₃, and there is essentially a unique maximal 2-dimensional permanental polytope of Ω₃ (a square of side $\text{1}\text{3}$). Permanental polytopes of dimension $\text{(n}{}^2\text{?3n+4)}\text{2}$ are exhibited for each n?4.     

A bound for the number of vertices of a polytope with applications

We prove that the number of vertices of a polytope of a particular kind is exponentially large in the dimension of the polytope. As a corollary, we prove that an n-dimensional centrally symmetric polytope with O(n) facets has {ie1-1} vertices and that the number of r-factors in a k-regular graph is exponentially large in the number of vertices of the graph provided k≥2r+1 and every cut in the graph with at least two vertices on each side has more than k/r edges.     

Ehrhart polynomials of cyclic polytopes

The Ehrhart polynomial of an integral convex polytope counts the number of lattice points in dilates of the polytope. In (Coefficients and roots of Ehrhart polynomials, preprint), the authors conjectured that for any cyclic polytope with integral parameters, the Ehrhart polynomial of it is equal to its volume plus the Ehrhart polynomial of its lower envelope and proved the case when the dimension d=2. In our article, we prove the conjecture for any dimension.     

The doubly stochastic matrices of a vector majorization

Let y be majorized by x. We investigate the polytope of doubly stochastic matrices D for which y = Dx. We determine when there is a positive D and when there is a fully indecomposable D. We calculate the dimension of the polytope, and as a result determine when D is uniquely determined.     

h-Vectors of Gorenstein polytopes

We show that the Ehrhart h-vector of an integer Gorenstein polytope with a regular unimodular triangulation satisfies McMullen's g-theorem; in particular, it is unimodal. This result generalizes a recent theorem of Athanasiadis (conjectured by Stanley) for compressed polytopes. It is derived from a more general theorem on Gorenstein affine normal monoids M: one can factor K[M] (K a field) by a “long” regular sequence in such a way that the quotient is still a normal affine monoid algebra. This technique reduces all questions about the Ehrhart h-vector of P to the Ehrhart h-vector of a Gorenstein polytope Q with exactly one interior lattice point, provided each lattice point in a multiple cP, c∈N, can be written as the sum of c lattice points in P. (Up to a translation, the polytope Q belongs to the class of reflexive polytopes considered in connection with mirror symmetry.) If P has a regular unimodular triangulation, then it follows readily that the Ehrhart h-vector of P coincides with the combinatorial h-vector of the boundary complex of a simplicial polytope, and the g-theorem applies.     

Classifying smooth lattice polytopes via toric fibrations

We show that any smooth Q-normal lattice polytope P of dimension n and degree d is a strict Cayley polytope if n?2d+1. This gives a sharp answer, for this class of polytopes, to a question raised by V.V. Batyrev and B. Nill.     

Bases-cobases graphs and polytopes of matroids

LetM be ablock matroid (i.e. a matroid whose ground setE is the disjoint union of two bases). We associate withM two objects:

1. Thebases-cobases graph G=G(M,M ^*) having as vertices the basesB ofM for which the complementE\B is also a base, and as edges the unordered pairs (B,B′) of such bases differing exactly by two elements.
2. Thepolytope of the bases-cobases K=K(M,M ^*) whose extreme points are the incidence vectors of the bases ofM whose complement is also a base.

We prove that, ifM is graphic (or cographic), the distance between any two vertices ofG corresponding to disjoint bases is equal to the rank ofM (generalizing a result of [10]). Concerning the polytope we prove thatK is an hypercube if and only if dim(K)=rank(M). A constructive characterization of the class of matroids realizing this equality is given.     

Method for polyhedral approximation of a ball with an optimal order of growth of the facet structure cardinality

The problem of polyhedral approximation of a multidimensional ball is considered. It is well known that the norm of the f-vector (the maximum number of faces of all dimensions) of an approximating polytope grows at least as fast as O(δ^{(1 ? d)/2}), where δ is the Hausdorff deviation and d is the space dimension. An iterative method, namely, the deep holes method is used to construct metric nets. As applied to the problem under study, the method sequentially supplements the vertex set of the polytope with its deep holes in the metric on the ball surface (i.e., with points of the surface that are farthest away from the vertices of the polytope). It is shown that the facet structure cardinality of the constructed polytope has an optimal growth rate. It is also shown that the number of faces of all dimensions in the approximating polytopes generated by the method is asymptotically proportional to the number of their vertices. Closed-form expressions for the constants are obtained, which depend only on the dimension of the space, including the case of high dimensions. For low dimensions (d ranging from 3 to 5), upper bounds for the growth rate of the number of faces of all dimensions are obtained depending on the accuracy of the approximation.     

Quasitoric Totally Normally Split Manifolds

A smooth stably complex manifold is called a totally tangentially/normally split manifold (TTS/TNS manifold for short) if the respective complex tangential/normal vector bundle is stably isomorphic to a Whitney sum of complex line bundles, respectively. In this paper we construct manifolds M such that any complex vector bundle over M is stably equivalent to a Whitney sum of complex line bundles. A quasitoric manifold shares this property if and only if it is a TNS manifold. We establish a new criterion for a quasitoric manifold M to be TNS via non-semidefiniteness of certain higher degree forms in the respective cohomology ring of M. In the family of quasitoric manifolds, this generalizes the theorem of J. Lannes about the signature of a simply connected stably complex TNS 4-manifold. We apply our criterion to show the flag property of the moment polytope for a nonsingular toric projective TNS manifold of complex dimension 3.     

Completely unimodal numberings of a simple polytope

A completely unimodal numbering of the m vertices of a simple d-dimensional polytope is a numbering 0, 1, …,m−1 of the vertices such that on every k-dimensional face (2≤k≤d) there is exactly one local minimum (a vertex with no lower-numbered neighbors on that face). Such numberings are abstract objective functions in the sense of Adler and Saigal [1]. It is shown that a completely unimodal numbering of the vertices of a simple polytope induces a shelling of the facets of the dual simplicial polytope. The h-vector of the dual simplicial polytope is interpreted in terms of the numbering (with respect to using a local-improvement algorithm to locate the vertex numbered 0). In the case that the polytope is combinatorially equivalent to a d-dimensional cube, a ‘successor-tuple’ for each vertex is defined which carries the crucial information of the numbering for local-improvement algorithms. Combinatorial properties of these d-tuples are studied. Finally the running time of one particular local-improvement algorithm, the Random Algorithm, is studied for completely unimodal numberings of the d-cube. It is shown that for a certain class of numberings (which includes the example of Klee and Minty [8] showing that the simplex algorithm is not polynomial and all Hamiltonian saddle-free injective pseudo-Boolean functions [6]) this algorithm has expected running time that is at worst quadratic in the dimension d.     

Fractional perfect b-matching polytopes I: General theory

The fractional perfect b-matching polytope of an undirected graph G is the polytope of all assignments of nonnegative real numbers to the edges of G such that the sum of the numbers over all edges incident to any vertex v   is a prescribed nonnegative number _bv$b_v$. General theorems which provide conditions for nonemptiness, give a formula for the dimension, and characterize the vertices, edges and face lattices of such polytopes are obtained. Many of these results are expressed in terms of certain spanning subgraphs of G which are associated with subsets or elements of the polytope. For example, it is shown that an element u of the fractional perfect b-matching polytope of G is a vertex of the polytope if and only if each component of the graph of u either is acyclic or else contains exactly one cycle with that cycle having odd length, where the graph of u is defined to be the spanning subgraph of G whose edges are those at which u is positive.     

Faces of faces of the acyclic Birkhoff polytope

Given a p-face of the acyclic Birkhoff polytope ?? _n (T), where T is a tree with n vertices, we find the number of faces of lower dimension that are contained in it, and its nature is discussed.     

A polyhedral study of triplet formulation for single row facility layout problem

The single row facility layout problem (SRFLP) is the problem of arranging n departments with given lengths on a straight line so as to minimize the total weighted distance between all department pairs. We present a polyhedral study of the triplet formulation of the SRFLP introduced by Amaral [A.R.S. Amaral, A new lower bound for the single row facility layout problem, Discrete Applied Mathematics 157 (1) (2009) 183-190]. For any number of departments n, we prove that the dimension of the triplet polytope is n(n−1)(n−2)/3 (this is also true for the projections of this polytope presented by Amaral). We then prove that several valid inequalities presented by Amaral for this polytope are facet-defining. These results provide theoretical support for the fact that the linear program solved over these valid inequalities gives the optimal solution for all instances studied by Amaral.     

A spectral approach to polyhedral dimension

Thespectrum spec( ) of a convex polytope is defined as the ordered (non-increasing) list of squared singular values of [A|1], where the rows ofA are the extreme points of . The number of non-zeros in spec( ) exceeds the dimension of by one. Hence, the dimension of a polytope can be established by determining its spectrum. Indeed, this provides a new method for establishing the dimension of a polytope, as the spectrum of a polytope can be established without appealing to a direct proof of its dimension. The spectrum is determined for the four families of polytopes defined as the convex hulls of:

1. The edge-incidence vectors of cutsets induced by balanced bipartitions of the vertices in the complete undirected graph on 2_q vertices (see Section 6).
2. The edge-incidence vectors of Hamiltonian tours in the complete undirected graph onn vertices (see Section 6).
3. The arc-incidence vectors of directed Hamiltonian tours in the complete directed graph ofn nodes (see Section 7).
4. The edge-incidence vectors of perfect matchings in the complete 3-uniform hypergraph on 3_q vertices (see Section 8).

In the cases of (ii) and (iii), the associated dimension results are well-known. The dimension results for (i) and (iv) do not seem to be well-known. General principles are discussed for ‘balanced polytopes’ arising from complete structures.     

Matroid Polytopes and their Volumes

We express the matroid polytope P _M of a matroid M as a signed Minkowski sum of simplices, and obtain a formula for the volume of P _M . This gives a combinatorial expression for the degree of an arbitrary torus orbit closure in the Grassmannian Gr _k,n . We then derive analogous results for the independent set polytope and the underlying flag matroid polytope of M. Our proofs are based on a natural extension of Postnikov’s theory of generalized permutohedra.     

On pedigree polytopes and Hamiltonian cycles

In this paper we define a combinatorial object called a pedigree, and study the corresponding polytope, called the pedigree polytope. Pedigrees are in one-to-one correspondence with the Hamiltonian cycles on K_n. Interestingly, the pedigree polytope seems to differ from the standard tour polytope, Q_n with respect to the complexity of testing whether two given vertices of the polytope are nonadjacent. A polynomial time algorithm is given for nonadjacency testing in the pedigree polytope, whereas the corresponding problem is known to be NP-complete for Q_n. We also discuss some properties of the pedigree polytope and illustrate with examples.     

Equidecomposable and weakly neighborly polytopes

A polytope is equidecomposable if all its triangulations have the same face numbers. For an equidecomposable polytope all minimal affine dependencies have an equal number of positive and negative coefficients. A subclass consists of the weakly neighborly polytopes, those for which every set of vertices is contained in a face of at most twice the dimension as the set. Theh-vector of every triangulation of a weakly neighborly polytope equals theh-vector of the polytope itself. Combinatorial properties of this class of polytopes are studied. Gale diagrams of weakly neighborly polytopes with few vertices are characterized in the spirit of the known Gale diagram characterization of Lawrence polytopes, a special class of weakly neighborly polytopes.     

The Flag Polynomial of the Minkowski Sum of Simplices

For a polytope we define the flag polynomial, a polynomial in commuting variables related to the well-known flag vector and describe how to express the flag polynomial of the Minkowski sum of k standard simplices in a direct and canonical way in terms of the k-th master polytope P(k) where ${k \in \mathbb {N}}$ . The flag polynomial facilitates many direct computations. To demonstrate this we provide two examples; we first derive a formula for the f -polynomial and the maximum number of d-dimensional faces of the Minkowski sum of two simplices. We then compute the maximum discrepancy between the number of (0, d)-chains of faces of a Minkowski sum of two simplices and the number of such chains of faces of a simple polytope of the same dimension and on the same number of vertices.     

Densities for random balanced sampling

A random balanced sample (RBS) is a multivariate distribution with n components X_k, each uniformly distributed on [-1,1], such that the sum of these components is precisely 0. The corresponding vectors lie in an (n-1)-dimensional polytope M(n). We present new methods for the construction of such RBS via densities over M(n) and these apply for arbitrary n. While simple densities had been known previously for small values of n (namely 2,3, and 4), for larger n the known distributions with large support were fractal distributions (with fractal dimension asymptotic to n as n→∞). Applications of RBS distributions include sampling with antithetic coupling to reduce variance, and the isolation of nonlinearities. We also show that the previously known densities (for n?4) are in fact the only solutions in a natural and very large class of potential RBS densities. This finding clarifies the need for new methods, such as those presented here.