On linear sections of lattice packings

Ind-dimensional euclidean spaceE ^d letP be a lattice packing of subsets ofE ^d , and letH be ak-dimensional linear subspace ofE ^d (0<k<d). Then,P induces a packing inH consisting of all setsPH withPP. The relationship between the density of this packing inH and the density ofP is investigated. A result from the theory of uniform distribution of linear forms is used to prove an integral formula that enables one to evaluate the density of the induced packing inH (under suitable assumptions on the sets ofP and the functionals used to define the densities). It is shown that this result leads to explicit formulas for the averages of the induced densities under the rotation ofH. If the densities are taken with respect to the mean cross-sectional measures of convex bodies one obtains analogues of the integral geometric intersection formulas of Crofton.Dedicated to Professor E. Hlawka on the occasion of his seventieth birthdaySupported by National Science Foundation Research Grant DMS 8300825.     

A new technique to compute polygonal schema for 2-manifolds with application to null-homotopy detection

We provide a new technique for deriving optimal-sized polygonal schema for triangulated compact 2-manifolds without boundary inO(n) time, wheren is the size of the given triangulationT. We first derive a polygonal schemaP embedded inT using Seifert-Van Kampen's theorem. A reduced polygonal schemaQ of optimal size is computed fromP, where a surjective map from the vertices ofP is retained to the vertices ofQ. This helps detecting null-homotopic (contractible to a point) cycles. Given a cycle of lengthk, we determine if it is null-homotopic inO(n+k logg) time and in θ(n+k) space, whereg is the genus of the given 2-manifold. The actual contraction for a null-homotopic cycle can be computed in θ(nk) time and space. This is a considerable improvement over the previous best-known algorithm for this problem.     

Generic determinantal schemes and the smoothability of determinantal schemes of codimension 2

The aim of this article is to prove a criterion for projectively Cohen-Macaulay two-codimensional subschemes ofP _k ^N to be smoothable. For curves inP _k ³ this criterion is due to T. Sauer [4]. The considered schemes are determinantal, so we study related generic affine determinantal schemes. We compute their dimension, and under the condition that the dimension is minimal we calculate the codimension of the singular locus.     

Extensions of invariant ideals

An ideai on a setX has the property S(k) iff for every partition ofX into sets of cardinality at leastk there exists a selector inI. We give a solution to a problem of Weglorz [6] proving, in particular, that if G is a group of uncountable, regular cardinalityk then every invariant,k-complete ideal on G can be extended to an invariant,k-complete ideal onG with the property S(k).Presented by Jan Mycielski.     

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.     

Path-closed sets

Given a digraphG = (V, E), call a node setT ⊆V path-closed ifv, v′ εT andw εV is on a path fromv tov′ impliesw εT. IfG is the comparability graph of a posetP, the path-closed sets ofG are the convex sets ofP. We characterize the convex hull of (the incidence vectors of) all path-closed sets ofG and its antiblocking polyhedron inR ^v , using lattice polyhedra, and give a minmax theorem on partitioning a given subset ofV into path-closed sets. We then derive good algorithms for the linear programs associated to the convex hull, solving the problem of finding a path-closed set of maximum weight sum, and prove another min-max result closely resembling Dilworth’s theorem.     

Path extendable graphs

A pathP in a graphG is said to beextendable if there exists a pathP’ inG with the same endvertices asP such thatV(P)⊆V (P’) and |V(P’)|=|V(P)|+1. A graphG ispath extendable if every nonhamiltonian path inG is extendable. We investigate the extent to which known sufficient conditions for a graph to be hamiltonian-connected imply the extendability of paths in the graph. Several theorems are proved: for example, it is shown that ifG is a graph of orderp in which the degree sum of each pair of non-adjacent vertices is at leastp+1 andP is a nonextendable path of orderk inG thenk≤(p+1)/2 and 〈V (P)〉≅K _k orK _k −e. As corollaries of this we deduce that if δ(G)≥(p+2)/2 or if the degree sum of each pair of nonadjacent vertices inG is at least (3p−3)/2 thenG is path extendable, which strengthen results of Williamson [13].     

On the dimension of ordered sets with the 2-cutset property

An ordered setP is said to have 2-cutset property if, for every elementx ofP, there is a setS of elements ofP which are noncomparable tox, with |S|?2, such that every maximal chain inP meets {x}∪S. We consider the following question: Does there exist ordered sets with the 2-cutset property which have arbitrarily large dimension? We answer the question in the negative by establishing the following two results.Theorem: There are positive integersc andd such that every ordered setP with the 2-cutset property can be represented asP=X∪Y, whereX is an ordinal sum of intervals ofP having dimension ?d, andY is a subset ofP having width ?c. Corollary: There is a positive integern such that every ordered set with the 2-cutset property has dimension ?n.     

Biclosure and bistability in a balanced bipartite graph

The k-biclosure of a balanced bipartite graph wiht color classes A and B is the graph obtained from G by recursively joining pairs of nonadjacent vertices respectively taken in A and B whose degree sum is at least k, until no such pair remains. A property P defined on all the balanced bipartite graphs of order 2n is k-bistable if whenever G + ab has property P and d_G(b) ≧ k then G itself has property P. We present a synthesis of results involving, for some properties, P, the bistability of P, the k-biclosure of G, the number of edges and the minimum degree. © 1995 John Wiley & Sons, Inc.     

Multiple-guard kernels of simple polygons

We study a generalization of the kernel of a polygon. A polygonP isk guardable if there arek points inP such that, for all pointsp inP, there is at least one of thek points that seesp. We call thek points ak-guard set ofP and thek-kernel of a polygonP is the union of allk-guard sets ofP. The usual definition of the kernel of a polygon is now the one-kernel in this notation.We show that the two-kernel of a simple polygonP withn edges has O(n⁴) components and that there are polygons whose two-kernel has (n) components. Moreover, we show that the components of the two-kernel of a simple polygon can be paired in a natural manner which implies that the two-kernel of a simple polygon has either one component or an even number of components. Finally, we consider the question of whether there is a non-star-shaped simple polygonP such that 2-kernel(P) = P. We show that when the two-kernel has only one component, it contains a hole; hence, the two-kernel of a simple polygonP is neverP itself. For everyk 1, there are, however, polygonsP _k with holes such thatk-kernel(P_k) = P_k.This work was supported under grant No. Ot 64/8-1, Diskrete Probleme from the the Deutsche Forschungsgemeinschaft, grants from the Natural Sciences and Engineering Council of Canada, from the Information Technology Research Centre of Ontario, and from the Research Grants Council of Hong Kong.     

Intersection properties of boxes. Part I: An upper-bound theorem

LetP be a family ofn boxes inR ^d (with edges parallel to the coordinate axes). Fork=0, 1, 2, …, denote byf _k (P) the number of subfamilies ofP of sizek+1 with non-empty intersection. We show that iff _r (P)=0 for somer≦n, then where thef _k (n, d, r) are ceg196rtain definite numbers defined by (3.4) below. The result is best possible for eachk. Fork=1 it was conjectured by G. Kalai (Israel J. Math.48 (1984), 161–174). As an application, we prove a ‘fractional’ Helly theorem for families of boxes inR ^d .     

Singular sets of convex bodies and surfaces with generalized curvatures

In this paper we consider generalized surfaces with curvature measures and we study the properties of those k-dimensional subsets Σ ^k of such surfaces where the curvatures have positive density with respect to k-dimensional Hausdorff measure. Special attention is given to boundaries of convex bodies inR ³. We introduce a class of convex sets whose curvatures live only on integer dimension sets. For such convex sets we consider integral functionals depending on the curvature and the area ofK and on the curvature andH ^k of Σ ^k .     

Toughness and Delaunay triangulations

We show that nondegenerate Delaunay triangulations satisfy a combinatorial property called 1-toughness. A graphG is1-tough if for any setP of vertices,c(G–P)|G|, wherec(G–P) is the number of components of the graph obtained by removingP and all attached edges fromG, and |G| is the number of vertices inG. This property arises in the study of Hamiltonian graphs: all Hamiltonian graphs are 1-tough, but not conversely. We also show that all Delaunay triangulationsT satisfy the following closely related property: for any setP of vertices the number of interior components ofT–P is at most |P|–2, where an interior component ofT–P is a component that contains no boundary vertex ofT. These appear to be the first nontrivial properties of a purely combinatorial nature to be established for Delaunay triangulations. We give examples to show that these bounds are best possible and are independent of one another. We also characterize the conditions under which a degenerate Delaunay triangulation can fail to be 1-tough. This characterization leads to a proof that all graphs that can be realized as polytopes inscribed in a sphere are 1-tough. One consequence of the toughness results is that all Delaunay triangulations and all inscribable graphs have perfect matchings.This research was sponsored in part by the National Science Foundation under Grant IRI-88-02457.     

Reduction theory for a rational function field

LetG be a split reductive group over a finite field F_q. LetF = F_q(t) and let A denote the adèles ofF. We show that every double coset inG(F)/G(A)/K has a representative in a maximal split torus ofG. HereK is the set of integral adèlic points ofG. WhenG ranges over general linear groups this is equivalent to the assertion that any algebraic vector bundle over the projective line is isomorphic to a direct sum of line bundles.     

On dimensional properties of graphs

A dimensional property of graphs is a propertyP such that every graphG is the intersection of graphs having propertyP. IfP is a dimensional property, we describe a general method for computing the least integerk so thatG is the intersection ofk graphs having propertyP. We give simple applications of the method to computing the boxicity, the cubicity, the circular dimension, the rigid circuit dimension, and the overlap dimension, and mention connections to other concepts such as the threshold dimension.     

On the Expected Number of Shadow Vertices of the Convex Hull of Random Points

Leta₁, . . . ,a_mbe independent random points in ⁿthat are independent and identically distributed spherically symmetrical in ⁿ. Moreover, letXbe the random polytope generated as the convex hull ofa₁, . . . ,a_mand letL_kbe an arbitraryk-dimensional subspace of ⁿwith 2 ≤k≤n− 1. LetX_kbe the orthogonal projection image ofXinL_k. We call those vertices ofXwhose projection images inL_kare vertices ofX_kshadow vertices ofXwith respect to the subspaceL_k. We derive a distribution independent sharp upper bound for the expected number of shadow vertices ofXinL_k.     

A generalized mean-value theorem

We give another proof of Seymour and Zaslavsky's theorem: For every familyf ₁,f ₂,...,f _n of continous functions defined on [0, 1], there exists a finite setF[0, 1] such that the average sum off _k overF coincides with the integral off _k for everyk=1, 2,...,n.     

Halfspace range search: An algorithmic application ofk-sets

Given a fixed setS ofn points inE ³ and a query plane, the halfspace range search problem asks for the retrieval of all points ofS on a chosen side of. We prove that withO(n(logn)⁸ (loglogn)⁴) storage it is possible to solve this problem inO(k+logn) time, wherek is the number of points to be reported. This result rests crucially on a new combinatorial derivation. We show that the total number ofj-sets (j=1, ...,k) realized by a set ofn points inE ³ isO(nk ⁵); ak-set is any subset ofS of sizek which can be separated from the rest ofS by a plane.Supported in part by NSF grants MCS 83-03925 and the Office of Naval Research and the Defense Advanced Research Projects Agency under contract N00014-83-K-0146 and ARPA Order No. 4786.Supported in part by Joint Services Electronics Program under Contract N00014-79-C-0424.     

The smallest degree sum that yields potentially Pk-graphical sequences

A simple graph G is said to have property P_k if it contains a complete subgraph of order k + 1, and a sequence π is potentially P_k-graphical if it has a realization having property P_k. Let σ (k, n) denote the smallest degree sum such that every n-term graphical sequence π without zero terms and with degree sum σ(π) ≥ σ(k, n) is potentially P_k-graphical. Erdós, Jacobson, and Lehel [Graph Theory, 1991, 439–449] conjectured that σ(k, n) = (k − 1)(2n − k) + 2. In this article, we prove that the conjecture is true for k = 4 and n ≥ 10. © 1998 John Wiley & Sons, Inc. J. Graph Theory 29: 63–72, 1998     

On minimax optimization problems

We give a short proof that in a convex minimax optimization problem ink dimensions there exist a subset ofk + 1 functions such that a solution to the minimax problem with thosek + 1 functions is a solution to the minimax problem with all functions. We show that convexity is necessary, and prove a similar theorem for stationary points when the functions are not necessarily convex but the gradient exists for each function.