Polytope Research Articles

Polytopal Linear Groups and Balanced 3-Polytopes

In [5] Bruns and Gubeladze have constructed Milnor K 2-groups for so-called balanced lattice polytopes and classified the balanced polytopes of dimension 2 up to equivalence of the stable elementary groups 𝔼(R, P). In [6] they have shown that the polytopes in a certain subclass, called Col-divisible, behave well for the construction of higher K-groups. In this article, we give a list of all possibly occurring E-equivalence classes of three-dimensional balanced polytopes. Finally, we compute 𝔼(R, P) provided P is a Col-divisible balanced polytope of dimension three.

Reflection Coefficients of Polynomials and Stable Polytopes

The geometry of stable discrete polynomials using their coefficients and reflection coefficients is investigated. Two linear Schur invariant transformations with a free parameter in the polynomial coefficient space are introduced. The first transformation Rfrn times RfrrarrRfrn maps an arbitrary stable polytope into another stable polytope. The second transformation Rfrn times RfrrarrRfrn maps a stable tilted n-dimensional hyperrectangle defined by the discrete Kharitonov theorem into a stable (n+1)- dimensional polytope.

On permutation polytopes

In this paper we lay the foundations for the study of permutation polytopes: the convex hull of a group of permutation matrices. We clarify the relevant notions of equivalence, prove a product theorem, and discuss centrally symmetric permutation polytopes. We provide a number of combinatorial properties of (faces of) permutation polytopes. As an application, we classify ⩽4-dimensional permutation polytopes and the corresponding permutation groups. Classification results and further examples are made available online. We conclude with several questions suggested by a general finiteness result.

Flow polytopes and the graph of reflexive polytopes

We suggest defining the structure of an unoriented graph R d on the set of reflexive polytopes of a fixed dimension d . The edges are induced by easy mutations of the polytopes to create the possibility of walks along connected components inside this graph. For this, we consider two types of mutations: Those provided by performing duality via nef-partitions, and those arising from varying the lattice. Then for d ≤ 3 , we identify the flow polytopes among the reflexive polytopes of each single component of the graph R d . For this, we present for any dimension d ≥ 2 an explicit finite list of quivers giving all d -dimensional reflexive flow polytopes up to lattice isomorphism. We deduce as an application that any such polytope has at most 6 ( d − 1 ) facets.

On Kalai’s Conjectures Concerning Centrally Symmetric Polytopes

In 1989, Kalai stated three conjectures A, B, C of increasing strength concerning face numbers of centrally symmetric convex polytopes. The weakest conjecture, A, became known as the “3d -conjectur...

MVA-BN HIV Polytope

MVA-BN HIV Polytope

Cook, Kannan and Schrijver’s example revisited

In 1990, Cook, Kannan and Schrijver [W. Cook, R. Kannan, A. Schrijver, Chvátal closures for mixed integer programming problems, Mathematical Programming 47 (1990) 155–174] proved that the split closure (the 1st 1-branch split closure) of a polyhedron is again a polyhedron. They also gave an example of a mixed-integer polytope in R 2 + 1 whose 1-branch split rank is infinite. We generalize this example to a family of high-dimensional polytopes and present a closed-form description of the k th 1-branch split closure of these polytopes for any k ≥ 1 . Despite the fact that the m -branch split rank of the ( m + 1 )-dimensional polytope in this family is 1, we show that the 2-branch split rank of the ( m + 1 )-dimensional polytope is infinite when m ≥ 3 . We conjecture that the t -branch split rank of the ( m + 1 )-dimensional polytope of the family is infinite for any 1 ≤ t ≤ m − 1 and m ≥ 2 .

Nonrational configurations, polytopes, and surfaces

This column is a place for those bits of contagious mathematics that travel from person to person in the community, because they are so elegant, surprising, or appealing that one has an urge to pass them on.

An illustrated theory of hyperbolic virtual polytopes

The paper gives an illustrated introduction to the theory of hyperbolic virtual polytopes and related counterexamples to A.D. Alexandrov’s conjecture.

Dedekind–Carlitz polynomials as lattice-point enumerators in rational polyhedra

We study higher-dimensional analogs of the Dedekind–Carlitz polynomials $${c(u,v;a,b): = \sum\limits_{k = 1}^{b - 1} {u^{\left[ {\frac{{ka}}{b}} \right]} } v^{k - 1}}$$ , where u and v are indeterminates and a and b are positive integers. Carlitz proved that these polynomials satisfy the reciprocity law $${(v - 1)\,{\rm c}\,(u, v; a, b) + (u - 1)\,{\rm c}\,(v, u; b, a) = u^{a-1} v^{b-1} - 1,}$$ from which one easily deduces many classical reciprocity theorems for the Dedekind sum and its generalizations. We illustrate that Dedekind–Carlitz polynomials appear naturally in generating functions of rational cones and use this fact to give geometric proofs of the Carlitz reciprocity law and various extensions of it. Our approach gives rise to new reciprocity theorems and computational complexity results for Dedekind–Carlitz polynomials, a characterization of Dedekind–Carlitz polynomials in terms of generating functions of lattice points in triangles, and a multivariate generalization of the Mordell–Pommersheim theorem on the appearance of Dedekind sums in Ehrhart polynomials of 3-dimensional lattice polytopes.

Simplex-inspired algorithms for solving a class of convex programming problems

We consider a class of convex programming problems whose objective function is given as a linear function plus a convex function whose arguments are linear functions of the decision variables and whose feasible region is a polytope. We show that there exists an optimal solution to this class of problems on a face of the constraint polytope of dimension not more than the number of arguments of the convex function. Based on this result, we develop a method to solve this problem that is inspired by the simplex method for linear programming. It is shown that this method terminates in a finite number of iterations in the special case that the convex function has only a single argument. We then use this insight to develop a second algorithm that solves the problem in a finite number of iterations for an arbitrary number of arguments in the convex function. A computational study illustrates the efficiency of the algorithm and suggests that the average-case performance of these algorithms is a polynomial of low order in the number of decision variables.

Strongly monotypic polytopes in ℝ3

AbstractMcMullen et al. [

Delannoy numbers and Legendre polytopes

We construct an $n$-dimensional polytope whose boundary complex is compressed and whose face numbers for any pulling triangulation are the coefficients of the powers of $(x-1)/2$ in the $n$-th Legendre polynomial. We show that the non-central Delannoy numbers count all faces in the lexicographic pulling triangulation that contain a point in a given open quadrant. We thus provide a geometric interpretation of a relation between the central Delannoy numbers and Legendre polynomials, observed over 50 years ago. The polytopes we construct are closely related to the root polytopes introduced by Gelfand, Graev, and Postnikov. \par No construisons un polytope de dimension $n$ dont le complexe de bord est comprimé et dont les nombres de faces dans toute triangulation "en tirant des sommets'' sont les coefficients des puissances de $(x-1)/2$ dans le $n$-ième polynôme de Legendre. Nous montrons que les nombres centraux de Delannoy comptent toutes les faces dans la triangulation "en tirant des sommets'' en ordre lexicographique qui contiennent un point dans un certain quadrant ouvert. Ainsi nous produisons une interprétation géométrique d'une relation entre les nombres de Delannoy centraux et les polynômes de Legendre, notée il y a 50 ans. Nos polytopes sont reliés intimement aux polytopes de racines introduits par Gelfand, Graev, et Postnikov.

On $d$-dimensional compact hyperbolic Coxeter polytopes with $d+4$ facets

We show that there is no compact hyperbolic Coxeter d-polytope with d+4 facets for d>7. This bound is sharp: examples of such polytopes up to dimension 7 were found by Bugaenko (1984). We also show that in dimension d=7 the polytope with 11 facets is unique.

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, $4\le n\le 7$. Combined with results of Esselmann this gives the classification of all compact hyperbolic Coxeter $n$-polytopes with $n+3$ facets, $n\ge 4$. Polytopes in dimensions $2$ and $3$ were classified by Poincaré and Andreev.

Zonotopes and four-dimensional superconformal field theories

The a-maximization technique proposed by Intriligator and Wecht allows us to determine the exact R-charges and scaling dimensions of the chiral operators of four-dimensional superconformal field theories. The problem of existence and uniqueness of the solution, however, has not been addressed in general setting. In this paper, it is shown that the a-function has always a unique critical point which is also a global maximum for a large class of quiver gauge theories specified by toric diagrams. Our proof is based on the observation that the a-function is given by the volume of a three dimensional polytope called "zonotope", and the uniqueness essentially follows from Brunn-Minkowski inequality for the volume of convex bodies. We also show a universal upper bound for the exact R-charges, and the monotonicity of a-function in the sense that a-function decreases whenever the toric diagram shrinks. The relationship between a-maximization and volume-minimization is also discussed.

Decomposability of Polytopes

Decomposability of Polytopes

Schläfli numbers and reduction formula

We define so-called poset-polynomials of a graded poset and use it to give an explicit and general description of the combinatorial numbers in Schläfli’s (combinatorial) reduction formula. For simplicial and simple polytopes these combinatorial numbers can be written as functions of the numbers of faces of the polytope and the tangent numbers. We use the constructed formulas to determine the volume of 4-dimensional Coxeter polytopes.

Multiples of Lattice Polytopes without Interior Lattice Points

Let $\Delta$ be an $n$-dimensional lattice polytope. The smallest non-negative integer $i$ such that $k \Delta$ contains no interior lattice points for $1 \leq k \leq n - i$ we call the degree of $\Delta$. We consider lattice polytopes of fixed degree $d$ and arbitrary dimension $n$. Our main result is a complete classification of $n$-dimensional lattice polytopes of degree $d=1$. This is a generalization of the classification of lattice polygons $(n=2)$ without interior lattice points due to Arkinstall, Khovanskii, Koelman and Schicho. Our classification shows that the secondary polytope of a lattice polytope of degree 1 is always a simple polytope.

Ji-Huan He's Ten Dimensional Polytope and High Energy Particle Physics

Ji-Huan He's Ten Dimensional Polytope and High Energy Particle Physics