On the covering index of convex bodies (original) (raw)

On the vertex index of convex bodies

Advances in Mathematics, 2007

We introduce the vertex index, vein(K), of a given centrally symmetric convex body K ⊂ R d , which, in a sense, measures how well K can be inscribed into a convex polytope with small number of vertices. This index is closely connected to the illumination parameter of a body, introduced earlier by the first named author, and, thus, related to the famous conjecture in Convex Geometry about covering of a d-dimensional body by 2 d smaller positively homothetic copies. We provide asymptotically sharp estimates (up to a logarithmic term) of this index in the general case. More precisely, we show that for every centrally symmetric convex body K ⊂ R d one has

Covering a Plane Convex Body by Its Negative Homothets

Journal of Geometry, 2009

For every positive integer k, let λ k denote the smallest positive number such that every plane convex body can be covered by k homothetic copies of itself with homothety ratio −λ k. In this note, we verify a conjecture of Januszewski and Lassak that λ7 = 10 17. Furthermore, we give an estimate for λ6.

Covering Convex Bodies by Cylinders and Lattice Points by Flats

Journal of Geometric Analysis, 2009

In connection with an unsolved problem of Bang (1951) we give a lower bound for the sum of the base volumes of cylinders covering a d-dimensional convex body in terms of the relevant basic measures of the given convex body. As an application we establish lower bounds on the number of k-dimensional flats (i.e. translates of k-dimensional linear subspaces) needed to cover all the integer points of a given convex body in d-dimensional Euclidean space for 1 ≤ k ≤ d − 1.

On the total perimeter of homothetic convex bodies in a convex container

Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, 2014

For two planar convex bodies, C and D, consider a packing S of n positive homothets of C contained in D. We estimate the total perimeter of the bodies in S, denoted per(S), in terms of per(D) and n. When all homothets of C touch the boundary of the container D, we show that either per(S) = O(log n) or per(S) = O(1), depending on how C and D "fit together," and these bounds are the best possible apart from the constant factors. Specifically, we establish an optimal bound per(S) = O(log n) unless D is a convex polygon and every side of D is parallel to a corresponding segment on the boundary of C (for short, D is parallel to C). When D is parallel to C but the homothets of C may lie anywhere in D, we show that per(S) = O((1+esc(S)) log n/ log log n), where esc(S) denotes the total distance of the bodies in S from the boundary of D. Apart from the constant factor, this bound is also the best possible.

The Geometry of Homothetic Covering and Illumination

Discrete Geometry and Symmetry

At a first glance, the problem of illuminating the boundary of a convex body by external light sources and the problem of covering a convex body by its smaller positive homothetic copies appear to be quite different. They are in fact two sides of the same coin and give rise to one of the important longstanding open problems in discrete geometry, namely, the Illumination Conjecture. In this paper, we survey the activity in the areas of discrete geometry, computational geometry and geometric analysis motivated by this conjecture. Special care is taken to include the recent advances that are not covered by the existing surveys. We also include some of our recent results related to these problems and describe two new approaches-one conventional and the other computer-assisted-to make progress on the illumination problem. Some open problems and conjectures are also presented.

Piercing Translates and Homothets of a Convex Body

Algorithmica, 2010

According to a classical result of Grünbaum, the transversal number τ (F) of any family F of pairwise-intersecting translates or homothets of a convex body C in R d is bounded by a function of d. Denote by α(C) (resp. β(C)) the supremum of the ratio of the transversal number τ (F) to the packing number ν(F) over all finite families F of translates (resp. homothets) of a convex body C in R d. Kim et al. recently showed that α(C) is bounded by a function of d for any convex body C in R d , and gave the first bounds on α(C) for convex bodies C in R d and on β(C) for convex bodies C in the plane. Here we show that β(C) is also bounded by a function of d for any convex body C in R d , and present new or improved bounds on both α(C) and β(C) for various convex bodies C in R d for all dimensions d. Our techniques explore interesting inequalities linking the covering and packing densities of a convex body. Our methods for obtaining upper bounds are constructive and lead to efficient constant-factor approximation algorithms for finding a minimum-cardinality point set that pierces a set of translates or homothets of a convex body.