Economical coverings of sets of lattice points

Lett(n, d) be the minimum numbert such that there aret of then ^d lattice points

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.     

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.     

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.

     

Large sets of coverings

Large sets of packings were investigated extensively. Much less is known about the dual problem, i.e., large sets of coverings. We examine two types of important questions in this context; what is the maximum number of disjoint optimal coverings? and what is the minimum number of optimal coverings for which the union covers the space? We give various constructions which give the optimal solutions and some good upper and lower bounds on both questions, respectively. © 1994 John Wiley & Sons, Inc.     

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.     

On finite lattice coverings

We consider finite lattice coverings of strictly convex bodies K. For planar centrally symmetric K we characterize the finite arrangements C _n such that conv , where C _n is a subset of a covering lattice for K (which satisfies some natural conditions). We prove that for a fixed lattice the optimal arrangement (measured with the parametric density) is either a sausage, a so-called double sausage or tends to a Wulff-shape, depending on the parameter. This shows that the Wulff-shape plays an important role for packings as well as for coverings. Further we give a version of this result for variable lattices. For the Euclidean d-ball we characterize the lattices, for which the optimal arrangement is a sausage, for large parameter. Received 19 May 1999.     

Representations of polytopes and polyhedral sets

In this paper we describe a variant of the diagram techniques, such as Gale diagrams for polytopes and positive diagrams for positive bases, which is appropriate for polyhedral sets. We obtain our new technique as a translation-invariant representation of polytopes or polyhedral sets. This approach leads naturally to simpler proofs of the familiar combinatorial diagram relationships. However, this method is more versatile than those previously employed, in that it can be used to investigate linear systems of polyhedral sets, and metrical properties such as volume. In particular, we give an easy proof of a result of Meyer on decomposability of polytopes, and a more perspicuous way of looking at the well-known theorem of Minkowski on the realizability of polytopes whose facets have given outer normal vectors and areas.     

Sums of moran sets

We find conditions on the dissection of Moran set so that the group it generates under addition has positive Lebesgue measure.     

SUMS OF MORAN SETS

We find conditions on the dissection of moran set so that the group it generates under addition has positive Lebesgue masure.     

Multiple lattice packings and coverings of spheres

Let be a lattice inR ⁿ . Consider the systemS of unit spheres centered at the lattice points of .S is called ak-fold lattice packing (covering) if each point inR ⁿ lies in at most (least)k of the open (closed) spheres ofS. Letd _k ⁿ (D _k ⁿ ) be the density of the closest (thinnest)k-fold lattice packing (covering) ofR ⁿ . After dealing several cases left by G. Fejes Tóth and A. Florian, we have concluded thatd _k ⁿ >kd ₁ ⁿ for all (n, k) (n2,k2) except (2, 2), (2, 3), (2, 4); andD _k ³ <k D ₁ ³ for allk2.     

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...     

Ramification points of double coverings of curves and Weierstrass points

Summary Let π: X→C be a double covering with X smooth curve and C elliptic curve. Let R(π)⊂X be the ramification locus of π. Every P∈R(π) is a Weierstrass point of X and we study the triples (C, π, X) for which the set of corresponding Weierstrass points have certain semigroups of non-gaps. We study the same problem also for triple cyclic coverings of C. Entrata in Redazione il 17 luglio 1998. The authors were partially supported by MURST and GNSAGA of CNR (Italy).