Solution of Hadwiger's Covering Problem for Centrally Symmetric Convex Bodies in E 3 (original) (raw)

On the Bezdek--Pach Conjecture for Centrally Symmetric Convex Bodies

Canadian Mathematical Bulletin, 2009

The Bezdek–Pach conjecture asserts that the maximum number of pairwise touching positive homothetic copies of a convex body in ℝ d is 2 d . Naszódi proved that the quantity in question is not larger than 2 d+1. We present an improvement to this result by proving the upper bound 3 · 2 d–1 for centrally symmetric bodies. Bezdek and Brass introduced the one-sided Hadwiger number of a convex body. We extend this definition, prove an upper bound on the resulting quantity, and show a connection with the problem of touching homothetic bodies.

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

On the covering index of convex bodies

Aequationes mathematicae, 2016

Covering a convex body by its homothets is a classical notion in discrete geometry that has resulted in a number of interesting and long-standing problems. Swanepoel introduced the covering parameter of a convex body as a means of quantifying its covering properties. In this paper, we introduce two relatives of the covering parameter called covering index and weak covering index, which upper bound well-studied quantities like the illumination number, the illumination parameter and the covering parameter of a convex body. Intuitively, the two indices measure how well a convex body can be covered by a relatively small number of homothets having the same relatively small homothety ratio. We show that the covering index is a lower semicontinuous functional on the Banach-Mazur space of convex bodies. We further show that the affine d-cubes minimize covering index in any dimension d, while circular disks maximize it in the plane. Furthermore, the covering index satisfies a nice compatibility with the operations of direct vector sum and vector sum. In fact, we obtain an exact formula for the covering index of a direct vector sum of convex bodies that works in infinitely many instances. This together with a minimization property can be used to determine the covering index of infinitely many convex bodies. As the name suggests, the weak covering index loses some of the important properties of the covering index. Finally, we obtain upper bounds on the covering and weak covering index.

The illumination conjecture and its extensions

Periodica Mathematica Hungarica, 2006

The Illumination Conjecture was raised independently by Boltyanski and Hadwiger in 1960. According to this conjecture any d-dimensional convex body can be illuminated by at most 2 d light sources. This is an important fundamental problem. The paper surveys the state of the art of the Illumination Conjecture. 1. The Illumination Conjecture Let K be a convex body (i.e. a compact convex set with nonempty interior) in the d-dimensional Euclidean space E d , d ≥ 2. According to Hadwiger [22] an exterior point p ∈ E d \ K of K illuminates the boundary point q of K if the half line emanating from p passing through q intersects the interior of K (at a point not between p and q). Furthermore, a family of exterior points of K say, p 1 , p 2 ,. .. , p n illuminates K if each boundary point of K is illuminated by at least one of the point sources p 1 , p 2 ,. .. , p n. Finally, the smallest n for which there exist n exterior points of K that illuminate K is called the illumination number of K Mathematics subject classification number: 52C17, 52B11.

A note on the illumination of convex bodies

Geometriae Dedicata, 1993

Let K c E d be a convex body and let lr(K ) denote the minimum number of rdimensional affine subspaces of E n lying outside K with which it is possible to illuminate K, where 0 _< r < d -1. We give a new proof of the theorem that I,(K) > I-(d + 1)/(r + 1)] with equality for smooth K.

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.