Lattice points in lattice polytopes

IfK is the underlying point-set of a simplicial complex of dimension at mostd whose vertices are lattice points, and ifG(K) is the number of lattice points inK, then the lattice point enumeratorG(K,t)=1+ _n1 G(nK)t ⁿ takes the formC(K, t)/(1–t) ^d+1, for some polynomialC(K, t). Here,C(K, t) is expressed as a sum of local terms, one for each face ofK. WhenK is a polytope or its boundary, there result inequalities between the numbersG _r (K), whereG(n K)= _r=0 ^d n ^r G _r (K).     

Notes on lattice points of zonotopes and lattice-face polytopes

Minkowski’s second theorem on successive minima gives an upper bound on the volume of a convex body in terms of its successive minima. We study the problem to generalize Minkowski’s bound by replacing the volume by the lattice point enumerator of a convex body. In this context we are interested in bounds on the coefficients of Ehrhart polynomials of lattice polytopes via the successive minima. Our results for lattice zonotopes and lattice-face polytopes imply, in particular, that for 0-symmetric lattice-face polytopes and lattice parallelepipeds the volume can be replaced by the lattice point enumerator.     

Strong wavefront lemma and counting lattice points in sectors

We compute the asymptotics of the number of integral quadratic forms with prescribed orthogonal decompositions and more generally, the asymptotics of the number of lattice points lying in sectors of affine symmetric spaces. A new key ingredient in this article is the strong wavefront lemma, which shows that the generalized Cartan decomposition associated to a symmetric space is uniformly Lipschitz.     

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.     

Residue formulae, vector partition functions and lattice points in rational polytopes

We obtain residue formulae for certain functions of several variables. As an application, we obtain closed formulae for vector partition functions and for their continuous analogs. They imply an Euler-MacLaurin summation formula for vector partition functions, and for rational convex polytopes as well: we express the sum of values of a polynomial function at all lattice points of a rational convex polytope in terms of the variation of the integral of the function over the deformed polytope.

     

Valuations on lattice polytopes

Let L be a lattice (that is, a Z-module of finite rank), and let L=P(L) denote the family of convex polytopes with vertices in L; here, convexity refers to the underlying rational vector space V=Q⊗L. In this paper it is shown that any valuation on L satisfies the inclusion-exclusion principle, in the strong sense that appropriate extension properties of the valuation hold. Indeed, the core result is that the class of a lattice polytope in the abstract group L=P(L) for valuations on L can be identified with its characteristic function in V. In fact, the same arguments are shown to apply to P(M), when M is a module of finite rank over an ordered ring, and more generally to appropriate families of (not necessarily bounded) polyhedra.     

Centrally symmetric polytopes with many faces

We present explicit constructions of centrally symmetric polytopes with many faces: (1) we construct a d-dimensional centrally symmetric polytope P with about 3 ^d/4 ≈ (1.316) ^d vertices such that every pair of non-antipodal vertices of P spans an edge of P, (2) for an integer k ≥ 2, we construct a d-dimensional centrally symmetric polytope P of an arbitrarily high dimension d and with an arbitrarily large number N of vertices such that for some 0 < δ _k < 1 at least (1 ? (δ _k ) ^d )( _k ^N ) k-subsets of the set of vertices span faces of P, and (3) for an integer k ≥ 2 and α > 0, we construct a centrally symmetric polytope Q with an arbitrarily large number of vertices N and of dimension d = k ^1+o(1) such that at least $(1 - k^{ - \alpha } )(_k^N )$ k-subsets of the set of vertices span faces of Q.     

Distance between vertices of lattice polytopes

Optimization Letters - A lattice (d,k)-polytope is the convex hull of a set of points in dimension d whose coordinates are integers ranging between 0 and k. We consider the largest possible...     

Classification of lattice-regular lattice convex polytopes

We completely describe lattice convex polytopes in ℝ ⁿ (for any dimension n) that are regular with respect to the group of affine transformations preserving the lattice. Supported in part by the RFBR (Grant Nos. SS-1972.2003.1 and 05-01-01012a) and the NWO-RFBR (Grant No. 047.011.2004.026/RFBR No. 05-02-89000-NWO_a).     

Partition polytopes over 1-dimensional points

Received March 1996 / Revised version received April 5, 1998 Published online January 20, 1999     

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.     

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.