Peter Gritzmann

Slices of L. Fejes Tóth's sausage conjecture

Let be k non-overlapping translates of the unit d -ball B d in euclidean d -space E d . Let C k denote the convex hull of their centres and let S k be a segment of length 2( k – 1). Furthermore, let V d denote the d -volume. L. Fejes Toth conjectured in [1], that, for d ≥ 5,

Estimates for the minimal width of polytopes inscribed in convex bodies

The paper deals with the problem of approximating point sets byn-point subsets with respect to the minimal widthw. Let, in particular, ℋd denote the family of all convex bodies in Euclideand-space, letA ⊂ ℋd and letn be an integer greater thand. Then we ask for the greatest number μ=Λn(A) such that everyA εA contains a polytope withn vertices which has minimal width at least μw(A). We give bounds for Λn(ℋd), for Λn(ℳ2133;d), and for Λn(Wd), where ℳ2133;d,Wd denote the families of centrally symmetric convex bodies and of bodies of constant width, respectively.

An application of valuation theory to two problems in discrete geometry

Abstract First a new approach is given to the technique of simple valuations previously introduced by McMullen. Then, this method is used to establish some cases of Wills conjecture on the number of lattice points in convex bodies and of L. Fejes Toths sausage-conjecture on finite packings of the unit ball.

On two finite covering problems of Bambah, Rogers, Woods and Zassenhaus

Bambah, Rogers, Woods, and Zassenhaus considered the general problem of covering planar convex bodiesC byk translates of a centrally-symmetric convex bodyK ofE2 with the ramification that these translates cover the convex hullCk of their centres. They proved interesting inequalities for the volume ofC andCk. In the present paper some analogous results in euclideand-spaceEd are given. It turns out that on one hand extremal configurations ford≥5 are of quite different type than in the planar case. On the other hand inequalities similar to the planar ones seem to exist in general. Inequalities in both directions for the volume and other quermass-integrals are given.

An upper estimate for the lattice point enumerator

Since Minkowski [29] gave his famous lattice point theorem for centrally symmetric convex bodies, a theorem that turned out to be of fundamental importance in number theory, much effort has been made to obtain tight estimates for the number of lattice points of a given lattice in convex bodies in terms of the basic quermass-integrals W o ,…, W d , whose eminent role shows in Hadwigers functional theorem [14, 15, 16, see also 17, p. 221–225]. (For the discrete analogues of W o ,…, W d see [2].) The present paper is concerned with an upper estimate of this kind.

Polyedrische 2-Mannigfaltigkeiten mit wenigen nicht-konvexen Ecken

LetP denote a polyhedral 2-manifold in ℝ3, i.e. a 2-dimensional cell-complex in ℝ3 whose underlying point-set is a closed connected 2-manifold. A vertexv ofP is called convex if at least one of the two components into whichP divides a sufficiently small ball centered atv is convex. It is shown that every polyhedral 2-manifold in ℝ3 of genusg>−1 contains at least five non-convex vertices and that for every positive integerg this bound is attained, i.e. there exists a polyhedral 2-manifold in ℝ3 of genusg with precisely five non-convex vertices.

Über diej-ten Überdeckungsdichten konvexer Körper

AbstractLet for positive integersj,k,d and convex bodiesK of Euclideand-spaceEd of dimension at leastj Vj, k (K) denote the maximum of the intrinsic volumesVj(C) of those convex bodies whosej-skeleton skelj(C) can be covered withk translates ofK. Then thej-thk-covering density θj,k (K) is the ratiok Vj(K)/Vj,k(K). In particular, θd,k refers to the case of covering the entire convex bodiesC and the density is measured with respect to the volume while forj=d-1 the surface of the bodiesC is covered and accordingly the density is measured with respect to the surface area.The paper gives the estimaten

Sausage-skin problems for finite coverings

1 leqslant theta _{j,k} (K)< e (j + sqrt {pi /2} sqrt {d - j} )< (d + 1) e