Convex Lattice Polytopes Research Articles

Convexity of Distinct Sum Sets

We study a combinatorial notion where given a set S of lattice points one takes the set of all sums of p distinct points in S, and we ask the question: ‘if S is the set of lattice points of a convex lattice polytope, is the resulting set also the set of lattice points of a convex lattice polytope?’ We obtain a positive result in dimension 2 and a negative result in higher dimensions. We apply this to the corner cut polyhedron.

Sectional genus and the volume of a lattice polytope

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 $$h^{*}$$ -vector. On the other hand, it is known that the sectional genus of a polarized variety has an upper bound, which is an extension of the Castelnuovo bound for the genus of a projective curve. In this paper, we prove the equivalence of these two bounds. Namely, a polarized toric variety has maximal sectional genus if and only if its associated polytope has minimal volume. This is a generalization of the known fact that polytopes corresponding to the anticanonical bundles of Gorenstein toric Fano varieties are reflexive polytopes (whose typical examples are minimal volume polytopes with only one interior lattice point).

Volumes of Convex Lattice Polytopes and A Question of V. I. Arnold

We show by a direct construction that there are at least exp\({\{cV^{(d-1)/(d+1)}\}}\) convex lattice polytopes in \({\mathbb{R}^d}\) of volume V that are different in the sense that none of them can be carried to an other one by a lattice preserving affine transformation. This is achieved by considering the family \({\mathcal{P}^{d}(r)}\) (to be defined in the text) of convex lattice polytopes whose volumes are between 0 and rd/d!. Namely we prove that for \({P \in \mathcal{P}^{d}(r), d!}\) vol P takes all possible integer values between crd–1 and rd where \({c > 0}\) is a constant depending only on d.

On a question of V. I. Arnol’d

Abstract We show by a construction that there are at least exp {cV(d−1)/(d+1)} convex lattice polytopes in ℝd of volume V that are different in the sense that none of them can be carried to an other one by a lattice preserving affine transformation.

THE BOUNDARY VOLUME OF A LATTICE POLYTOPE

AbstractFor a d-dimensional convex lattice polytope P, a formula for the boundary volume vol(∂P) is derived in terms of the number of boundary lattice points on the first ⌊d/2⌋ dilations of P. As an application we give a necessary and sufficient condition for a polytope to be reflexive, and derive formulas for the f-vector of a smooth polytope in dimensions three, four, and five. We also give applications to reflexive order polytopes, and to the Birkhoff polytope.

On the classification of convex lattice polytopes

Abstract In 1980, Arnold studied the classification problem for convex lattice polygons of given area. Since then this problem and its high dimensional analogue have been studied by Bárány, Pach, Vershik and others. Bounds for the number of non-equivalent d-dimensional convex lattice polytopes of given volume have been achieved. In this paper we study Arnold's problem for centrally symmetric lattice polygons and the classification problem for convex lattice polytopes of given cardinality. In the plane we obtain analogues to the bounds of Arnold, Bárány and Pach in both cases. However, the number of non-equivalent d-dimensional convex lattice polytopes of w lattice points is infinite whenever w – 1 ≥ d ≥ 3, which may intuitively contradict to Bárány and Vershik's upper bound.

Lower bounds on the coefficients of Ehrhart polynomials

We present lower bounds for the coefficients of Ehrhart polynomials of convex lattice polytopes in terms of their volume. Concerning the coefficients of the Ehrhart series of a lattice polytope we show that Hibi’s lower bound is not true for lattice polytopes without interior lattice points. The counterexample is based on a formula of the Ehrhart series of the join of two lattice polytope. We also present a formula for calculating the Ehrhart series of integral dilates of a polytope.

Ehrhart Polynomials and Successive Minima

We investigate the Ehrhart polynomial for the class of 0-symmetric convex lattice polytopes in Euclidean $n$-space $\mathbb{R}^n$. It turns out that the roots of the Ehrhart polynomial and Minkowski's successive minima are closely related by their geometric and arithmetic mean. We also show that the roots of lattice $n$-polytopes with or without interior lattice points differ essentially. Furthermore, we study the structure of the roots in the planar case. Here it turns out that their distribution reflects basic properties of lattice polygons.

Viro theorem and topology of real and complex combinatorial hypersurfaces

We introduce a class of combinatorial hypersurfaces in the complex projective space. They are submanifolds of codimension 2 inℂP n and are topologically “glued” out of algebraic hypersurfaces in (ℂ*) n . Our construction can be viewed as a version of the Viro gluing theorem relating topology of algebraic hypersurfaces to the combinatorics of subdivisions of convex lattice polytopes. If a subdivision is convex, then according to the Viro theorem a combinatorial hypersurface is isotopic to an algebraic one. We study combinatorial hypersurfaces resulting from non-convex subdivisions of convex polytopes, show that they are almost complex varieties, and in the real case, they satisfy the same topological restrictions (congruences, inequalities etc.) as real algebraic hypersurfaces.

Counting lattice points by means of the residue theorem

We use the residue theorem to derive an expression for the number of lattice points in a dilated n-dimensional tetrahedron with vertices at lattice points on each coordinate axis and the origin. This expression is known as the Ehrhart polynomial. We show that it is a polynomial in t, where t is the integral dilation parameter. We prove the Ehrhart-Macdonald reciprocity law for these tetrahedra, relating the Ehrhart polynomials of the interior and the closure of the tetrahedra. To illustrate our method, we compute the Ehrhart coefficient for codimension 2. Finally, we show how our ideas can be used to compute the Ehrhart polynomial for an arbitrary convex lattice polytope.

The Ehrhart Polynomial of a Lattice Polytope

The problem of counting the number of lattice points inside a convex lattice polytope in R' (a polytope whose vertices have integer coordinates) has been studied from a variety of perspectives. Here we use the Poisson summation formula and related techniques in Fourier analysis to obtain a formula for the number of lattice points inside all integral dilates of a lattice polytope. We show that an associated generating function has an explicit representation in terms of sums of cotangents, which are higher-dimensional generalizations of Dedekind sums. Let En denote the n-dimensional integer lattice in R', and let P be an n-dimensional lattice polytope in R', which is a compact simplicial complex of pure dimension n whose vertices lie on the lattice. Consider the function of an integer-valued variable t that describes the number of lattice points that lie inside the dilated polytope tP:

Lefschetz Fixed-Point Theorem and Lattice Points in Convex Polytopes

A simple convex lattice polytope □ defines a torus-equivalent line bundle L□ over a toric variety X□ whose cohomology is intimately related to the lattice points of □. This is the starting point for much work on lattice points by algebrogeometric methods. After an introduction to toric geometry, a generalization of Atiyah and Bott′s Lefschetz fixed-point theorem is applied to the torus action on L□ to obtain information about the lattice points of □. In particular an explicit formula is derived, computing the number of lattice points and the volume of □ in terms of Todd polynomials in the geometric data at its extreme points. An alternative formulation of results of Brion [6] is derived and an elementary convex geometric interpretation is given by extending Ishida′s [12] Laurent expansions.

Computing the Ehrhart polynomial of a convex lattice polytope

We prove that computation of any fixed number of highest coefficients of the Ehrhart polynomial of an integral polytope can be reduced in polynomial time to computation of the volumes of faces.

On the number of convex lattice polytopes

On the number of convex lattice polytopes