On Extensive Subsets of Convex Bodies

A subset S of a d-dimensional convex body K is extensive if S ∂K and for any p, q ∈ S the distance between p and q is at least one-half of the maximum length of chords of K parallel to the segment pq. In this paper we establish the general upper bound |S| ≤ 3 ^d — 1. We also find an upper bound for a certain class of 3-polytopes, which leads to the determination of the maximum cardinalities of extensive subsets and their extremal configurations for tetrahedra, octahedra and some other 3-polytopes. This revised version was published online in June 2006 with corrections to the Cover Date.     

Exponential Lower Bound for the Translative Kissing Numbers of d -Dimensional Convex Bodies

The translative kissing number H(K) of a d -dimensional convex body K is the maximum number of mutually nonoverlapping translates of K which touch K . In this paper we show that there exists an absolute constant c > 0 such that H(K)≥ 2 ^cd for every positive integer d and every d -dimensional convex body K . We also prove a generalization of this result for pairs of centrally symmetric convex bodies. <lsiheader> <onlinepub>26 June, 1998 <editor>Editors-in-Chief: &lsilt;a href=../edboard.html#chiefs&lsigt;Jacob E. Goodman, Richard Pollack&lsilt;/a&lsigt; <pdfname>19n3p447.pdf <pdfexist>yes <htmlexist>no <htmlfexist>no <texexist>yes <sectionname> </lsiheader> Received February 18, 1997, and in revised form April 15, 1997.     

On Translational Clouds for a Convex Body

For a d-dimensional convex body K let C(K) denote the minimum size of translational clouds for K. That is, C(K) is the minimum number of mutually non-overlapping translates of K which do not overlap K and block all the light rays emanating from any point of K. In this paper we prove the general upper bound . Furthermore, for an arbitrary centrally symmetric d-dimensional convex body S we show . Finally, for the d-dimensional ball B^d we obtain the bounds .     

A Lower Bound for the Translative Kissing Numbers of Simplices

H (K) of a d-dimensional convex body K is the maximum number of mutually non-overlapping translates of K that can be arranged so that all touch K. In this paper we show that holds for any d-dimensional simplex (). We also prove similar inequalities for some, more general classes of convex bodies. Received May 18, 1998     

Model selection via information criteria

Summary This is a survey of the information criterion approach to model selection problems. New results about context tree estimation and the estimation of the basic neighborhood of Markov random fields are also mentioned.     

On a Lemma of Minkowski

In this paper we show that a generalization of a lemma of Minkowski can be applied to solve two problems concerning kissing numbers of convex bodies. In the first application we give a short proof showing that the lattice kissing number of tetrahedra is eighteen. Moreover, it turns out that for any tetrahedron T there exist a unique 18-neighbour lattice packing of T and an essentially unique 16-neighbour lattice packing of T. Secondly we show that for every integer d ≥ 3 there exists a d-dimensional convex body K such that the difference between its translative kissing number and lattice kissing number is at least 2^d-1.     

Ball and spindle convexity with respect to a convex body

Let ${C \subset \mathbb{R}^n}$ be a convex body. We introduce two notions of convexity associated to C. A set K is C-ball convex if it is the intersection of translates of C, or it is either ${\emptyset}$ , or ${\mathbb{R}^n}$ . The C-ball convex hull of two points is called a C-spindle. K is C-spindle convex if it contains the C-spindle of any pair of its points. We investigate how some fundamental properties of conventional convex sets can be adapted to C-spindle convex and C-ball convex sets. We study separation properties and Carathéodory numbers of these two convexity structures. We investigate the basic properties of arc-distance, a quantity defined by a centrally symmetric planar disc C, which is the length of an arc of a translate of C, measured in the C-norm that connects two points. Then we characterize those n-dimensional convex bodies C for which every C-ball convex set is the C-ball convex hull of finitely many points. Finally, we obtain a stability result concerning covering numbers of some C-ball convex sets, and diametrically maximal sets in n-dimensional Minkowski spaces.     

Solution of Hadwiger-Levi's Covering Problem for Duals of Cyclic 2k-Polytopes

For a collection C of convex bodies let h(C) be the minimum number m with the property that every element K of C can be covered by m or fewer smaller homothetic copies of K. Denote by C _d ^* the collection of all duals of cyclic d-polytopes, d 2. We show that h(C _2k ^* )=(k +1)2 for any k 2. We also prove the inequalities (d+1) (d+3)/4 h(C _d ^* ) (d+1) 2/2$ for any d 2.