Facets with fewest vertices

Forv>d≧3, letm(v, d) be the smallest numberm, such that every convexd-polytope withv vertices has a facet with at mostm vertices. In this paper, bounds form(v, d) are found; in particular, for fixedd≧3, $$\frac{{r - 1}}{r} \leqslant \mathop {\lim \inf }\limits_{\upsilon \to \infty } \frac{{m(\upsilon ,d)}}{\upsilon } \leqslant \mathop {\lim \sup }\limits_{\upsilon \to \infty } \frac{{m(\upsilon ,d)}}{\upsilon } \leqslant \frac{{d - 3}}{{d - 2}}$$ , wherer=[1/3(d+1)].     

On the second smallest distance between finitely many points on the sphere

In the following note we investigate the second smallest distance between finitely many points on the sphere. Actually we look for the smallest upper bound for the second smallest distance between n points on the unit sphere. We solve this problem for n=9 and also we give a general, non-trivial upper bound for the second smallest distance of n points with n9.Supported by the Hungarian National Foundation for Scientific Research, Number 1238.     

A note on the illumination of convex bodies

LetKE ^d be a convex body and letl _r(K) denote the minimum number ofr-dimensional affine subspaces ofE ^d lying outsideK with which it is possible to illuminateK, where 0rd–1. We give a new proof of the theorem thatl _r(K)(d+1)/(r+1) with equality for smoothK.The work was supported by Hung. Nat. Found. for Sci . Research No. 326-0213 and 326-0113.     

Grouped coordinate minimization using Newton's method for inexact minimization in one vector coordinate

Letf(x,y) be a function of the vector variablesx R ⁿ andy R ^m. The grouped (variable) coordinate minimization (GCM) method for minimizingf consists of alternating exact minimizations in either of the two vector variables, while holding the other fixed at the most recent value. This scheme is known to be locally,q-linearly convergent, and is most useful in certain types of statistical and pattern recognition problems where the necessary coordinate minimizers are available explicitly. In some important cases, the exact minimizer in one of the vector variables is not explicitly available, so that an iterative technique such as Newton's method must be employed. The main result proved here shows that a single iteration of Newton's method solves the coordinate minimization problem sufficiently well to preserve the overall rate of convergence of the GCM sequence.The authors are indebted to Professor R. A. Tapia for his help in improving this paper.     

On the (n − 2)-transversals of n convex subsets of the plane

In this paper we consider families of distinct ovals in the plane, with the property that certain subfamilies have stabbing lines (transversals). Our main result says that if any k member of the family can be stabbed by a line avoiding all the other ovals and k is large enough, then the family consists of at most k+1 ovals. For any n4 we show a family of n ovals, whose n–2 element subfamilies have, but the n–1 element subfamilies do not have, transversals.     

Maximum density space packing with parallel strings of spheres

A string of spheres is a sequence of nonoverlapping unit spheres inR ³ whose centers are collinear and such that each sphere is tangent to exactly two other spheres. We prove that if a packing with spheres inR ³ consists of parallel translates of a string of spheres, then the density of the packing is smaller than or equal to . This density is attained in the well-known densest lattice sphere packing. A long-standing conjecture is that this density is maximum among all sphere packings in space, to which our proof can be considered a partial result. The work of A. Bezdek and E. Makai was partially supported by the Hungarian National Foundation for Scientific Research under Grant Number 1238.     

Improving Rogers' Upper Bound for the Density of Unit Ball Packings via Estimating the Surface Area of Voronoi Cells from Below in Euclidean \sl d -Space for All \sl d ≥ \bf 8

   Abstract. The sphere packing problem asks for the densest packing of unit balls in E ^d . This problem has its roots in geometry, number theory and information theory and it is part of Hilbert's 18th problem. One of the most attractive results on the sphere packing problem was proved by Rogers in 1958. It can be phrased as follows. Take a regular d -dimensional simplex of edge length 2 in E ^d and then draw a d -dimensional unit ball around each vertex of the simplex. Let σ _d denote the ratio of the volume of the portion of the simplex covered by balls to the volume of the simplex. Then the volume of any Voronoi cell in a packing of unit balls in E ^d is at least ω _d /σ _d , where ω _d denotes the volume of a d -dimensional unit ball. This has the immediate corollary that the density of any unit ball packing in E ^d is at most σ _d . In 1978 Kabatjanskii and Levenštein improved this bound for large d . In fact, Rogers' bound is the presently known best bound for 4≤ d≤ 42 , and above that the Kabatjanskii—Levenštein bound takes over. In this paper we improve Rogers' upper bound for the density of unit ball packings in Euclidean d -space for all d≥ 8 and improve the Kabatjanskii—Levenštein upper bound in small dimensions. Namely, we show that the volume of any Voronoi cell in a packing of unit balls in E ^d , d≥ 8 , is at least ω _d /

4th GEOMETRY FESTIVAL,BUDAPEST

Periodica Mathematica Hungarica -