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Karoly Bezdek

Publications and source records attributed to Karoly Bezdek.

26 records · Page 2Linked to original sources
arXiv2011

On the vertex index of convex bodies

We introduce the vertex index, vein(K), of a given centrally symmetric convex body K, 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. Also, we provide sharp estimates in dimensions 2 and 3.

math.MG↗
arXiv2011

Contact numbers for congruent sphere packings in Euclidean 3-space

Continuing the investigations of Harborth (1974) and the author (2002) we study the following two rather basic problems on sphere packings. Recall that the contact graph of an arbitrary finite packing of unit balls (i.e., of an arbitrary finite family of non-overlapping unit balls) in Euclidean 3-space is the (simple) graph whose vertices correspond to the packing elements and whose two vertices are connected by an edge if the corresponding two packing elements touch each other. One of the most basic questions on contact graphs is to find the maximum number of edges that a contact graph of a packing of n unit balls can have in Euclidean 3-space. Our method for finding lower and upper estimates for the largest contact numbers is a combination of analytic and combinatorial ideas and it is also based on some recent results on sphere packings. Finally, we are interested also in the following more special version of the above problem. Namely, let us imagine that we are given a lattice unit sphere packing with the center points forming the lattice L in Euclidean 3-space (and with certain pairs of unit balls touching each other) and then let us generate packings of n unit balls such that each and every center of the n unit balls is chosen from L. Just as in the general case we are interested in finding good estimates for the largest contact number of the packings of n unit balls obtained in this way.

math.MG↗
arXiv2011

Illuminating spindle convex bodies and minimizing the volume of spherical sets of constant width

A subset of the d-dimensional Euclidean space having nonempty interior is called a spindle convex body if it is the intersection of (finitely or infinitely many) congruent d-dimensional closed balls. The spindle convex body is called a "fat" one, if it contains the centers of its generating balls. The core part of this paper is an extension of Schramm's theorem and its proof on illuminating convex bodies of constant width to the family of "fat" spindle convex bodies.

math.MG↗
arXiv2009

Covering convex bodies by cylinders and lattice points by flats

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 0<k<d.

math.MG↗
arXiv2009

From the Kneser-Poulsen conjecture to ball-polyhedra

A very fundamental geometric problem on finite systems of spheres was independently phrased by Kneser (1955) and Poulsen (1954). According to their well-known conjecture if a finite set of balls in Euclidean space is repositioned so that the distance between the centers of every pair of balls is decreased, then the volume of the union (resp., intersection) of the balls is decreased (resp., increased). In the first half of this paper we survey the state of the art of the Kneser-Poulsen conjecture in Euclidean, spherical as well as hyperbolic spaces with the emphases being on the Euclidean case. Based on that it seems very natural and important to study the geometry of intersections of finitely many congruent balls from the viewpoint of discrete geometry in Euclidean space. We call these sets ball-polyhedra. In the second half of this paper we survey a selection of fundamental results known on ball-polyhedra. Besides the obvious survey character of this paper we want to emphasize our definite intention to raise quite a number of open problems to motivate further research.

math.MG↗
arXiv2009

Tarski's plank problem revisited

In the 1930's, Tarski introduced his plank problem at a time when the field Discrete Geometry was about to born. It is quite remarkable that Tarski's question and its variants continue to generate interest in the geometric and analytic aspects of coverings by planks in the present time as well. The paper is a survey type with a list of open research problems.

math.MG↗
arXiv2001

Improving Rogers' upper bound for the density of unit ball packings via estimating the surface area of Voronoi cells from below in Euclidean d-space for all d>7

The sphere packing problem asks for the densest packing of unit balls in d-dimensional Euclidean space. This problem has its roots in geometry, number theory and it is part of Hilbert's 18th problem. In 1958 C. A. Rogers proved a non-trivial upper bound for the density of unit ball packings in d-dimensional Euclidean space for all d>0. In 1978 Kabatjanskii and Levenstein improved this bound for large d. In fact, Rogers' bound is the presently known best bound for 43>d>3, and above that the Kabatjanskii-Levenstein 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>7. We do this by estimating from below the surface area of Voronoi cells in any packing of unit balls in Euclidean d-space for all d>7.

math.MG↗