Facial structures of lattice path matroid polytopes

A lattice path matroid is a transversal matroid corresponding to a pair of lattice paths on the plane. A matroid base polytope is the polytope whose vertices are the incidence vectors of the bases of the given matroid. In this paper, we study the facial structures of matroid base polytopes corresponding to lattice path matroids. In the case of a border strip, we show that all faces of a lattice path matroid polytope can be described by certain subsets of deletions, contractions, and direct sums. In particular, we express them in terms of the lattice path obtained from the border strip. Subsequently, the facial structures of a lattice path matroid polytope for a general case are explained in terms of certain tilings of skew shapes inside the given region.     

Embedding a polytope in a lattice

We present a special similarity ofR ⁴ⁿ which maps lattice points into lattice points. Applying this similarity, we prove that if a (4n−1)-polytope is similar to a lattice polytope (a polytope whose vertices are all lattice points) inR ⁴ⁿ , then it is similar to a lattice polytope inR ⁴ⁿ⁻¹, generalizing a result of Schoenberg [4]. We also prove that ann-polytope is similar to a lattice polytope in someR ^N if and only if it is similar to a lattice polytope inR ²ⁿ⁺¹, and if and only if sin²(<ABC) is rational for any three verticesA, B, C of the polytope.     

The Contact Polytope of the Leech Lattice

The contact polytope of a lattice is the convex hull of its shortest vectors. In this paper we classify the facets of the contact polytope of the Leech lattice up to symmetry. There are 1,197,362,269,604,214,277,200 many facets in 232 orbits.     

Zonotopes whose cellular strings are all coherent

A cellular string of a polytope is a sequence of faces stacked on top of each other in a given direction. The poset of cellular strings, ordered by refinement, is known to be homotopy equivalent to a sphere. The subposet of coherent cellular strings is the face lattice of the fiber polytope, hence is homeomorphic to a sphere. In some special cases, every cellular string is coherent. Such polytopes are said to be all-coherent. We give a complete classification of zonotopes with the all-coherence property in terms of their oriented matroid structure. Although the face lattice of the fiber polytope in this case is not an oriented matroid invariant, we prove that the all-coherence property is invariant.     

Lattices generated by skeletons of reflexive polytopes

Lattices generated by lattice points in skeletons of reflexive polytopes are essential in determining the fundamental group and integral cohomology of Calabi-Yau hypersurfaces. Here we prove that the lattice generated by all lattice points in a reflexive polytope is already generated by lattice points in codimension two faces. This answers a question of John Morgan.     

A characterization of state spaces of orthomodular lattices

It is shown that every rational polytope is affinely equivalent to the set of all states of a finite orthomodular lattice, and that every compact convex subset of a locally convex topological vector space is affinely homeomorphic to the set of all states of an orthomodular lattice.     

Higher integrality conditions, volumes and Ehrhart polynomials

A polytope is integral if all of its vertices are lattice points. The constant term of the Ehrhart polynomial of an integral polytope is known to be 1. In previous work, we showed that the coefficients of the Ehrhart polynomial of a lattice-face polytope are volumes of projections of the polytope. We generalize both results by introducing a notion of k-integral polytopes, where 0-integral is equivalent to integral. We show that the Ehrhart polynomial of a k-integral polytope P has the properties that the coefficients in degrees less than or equal to k are determined by a projection of P, and the coefficients in higher degrees are determined by slices of P. A key step of the proof is that under certain generality conditions, the volume of a polytope is equal to the sum of volumes of slices of the polytope.     

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.     

Sectional genus and the volume of a lattice polytope

Journal of Algebraic Combinatorics - For a convex lattice polytope having at least one interior lattice point, a lower bound for its volume is derived from Hibi’s lower bound theorem for the...     

整数规划中的正则分解

In this paper, we study the problem of regular decomposition in integer programming. We apply the radical of binomial ideal and universal Gr¨obner bases to get the regular decomposition forms of a finite integer lattice point set. We indicate the relationship between state polytope and regular decompositions, i.e., an edge of state polytope corresponds to a binomial which decides one of regular decomposition forms of a finite integer lattice point set.     

On Smooth Lattice Polytopes with Small Degree

Toric geometry provides a bridge between the theory of polytopes and algebraic geometry: one can associate to each lattice polytope a polarized toric variety. In this article, we explore this correspondence to classify smooth lattice polytopes having small degree, extending a classification provided by Dickenstein, Di Rocco, and Piene. We follow their approach of interpreting the degree of a polytope as a geometric invariant of the corresponding polarized variety, and then apply techniques from Adjunction Theory and Mori Theory.     

Gorenstein polytopes and their stringy E-functions

Inspired by ideas from algebraic geometry, Batyrev and the first named author have introduced the stringy E-function of a Gorenstein polytope. We prove that this a priori rational function is actually a polynomial, which is part of a conjecture of Batyrev and the first named author. The proof relies on a comparison result for the lattice point structure of a Gorenstein polytope P, a face F of P and the face of the dual Gorenstein polytope corresponding to F. In addition, we study joins of Gorenstein polytopes and introduce the notion of an irreducible Gorenstein polytope. We show how these concepts relate to the decomposition of nef-partitions.     

Lattice polytopes of degree 2

A theorem of Scott gives an upper bound for the normalized volume of lattice polygons with exactly i>0 interior lattice points. We will show that the same bound is true for the normalized volume of lattice polytopes of degree 2 even in higher dimensions. In particular, there is only a finite number of quadratic polynomials with fixed leading coefficient being the h^∗-polynomial of a lattice polytope.     

Sums of sets of lattice points and unimodular coverings of polytopes

If P is a lattice polytope (that is, the convex hull of a finite set of lattice points in \({\mathbf{R}^n}\)), then every sum of h lattice points in P is a lattice point in the h-fold sumset hP. However, a lattice point in the h-fold sumset hP is not necessarily the sum of h lattice points in P. It is proved that if the polytope P is a union of unimodular simplices, then every lattice point in the h-fold sumset hP is the sum of h lattice points in P.     

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.     

Extensions of regular polytopes with preassigned Schläfli symbol

We say that a (d+1)-polytope P is an extension of a polytope K if the facets or the vertex figures of P are isomorphic to K. The Schläfli symbol of any regular extension of a regular polytope is determined except for its first or last entry. For any regular polytope K we construct regular extensions with any even number as first entry of the Schläfli symbol. These extensions are lattices if K is a lattice. Moreover, using the so-called CPR graphs we provide a more general way of constructing extensions of polytopes.     

The many aspects of counting lattice points in polytopes

A wide variety of topics in pure and applied mathematics involve the problem of counting the number of lattice points inside a convex bounded polyhedron, for short called a polytope. Applications range from the very pure (number theory, toric Hilbert functions, Kostant’s partition function in representation theory) to the most applied (cryptography, integer programming, contingency tables). This paper is a survey of this problem and its applications. We review the basic structure theorems about this type of counting problem. Perhaps the most famous special case is the theory of Ehrhart polynomials, introduced in the 1960s by Eugène Ehrhart. These polynomials count the number of lattice points in the different integral dilations of an integral convex polytope. We discuss recent algorithmic solutions to this problem and conclude with a look at what happens when trying to count lattice points in more complicated regions of space.