arXiv subjects

Károly Bezdek

Publications and source records attributed to Károly Bezdek.

23 records · Page 2Linked to original sources

arXiv2018

On contact graphs of totally separable packings in low dimensions

The contact graph of a packing of translates of a convex body in Euclidean $d$-space $\mathbb E^d$ is the simple graph whose vertices are the members of the packing, and whose two vertices are connected by an edge if the two members touch each other. A packing of translates of a convex body is called totally separable, if any two members can be separated by a hyperplane in $\mathbb E^d$ disjoint from the interior of every packing element. We give upper bounds on the maximum vertex degree (called separable Hadwiger number) and the maximum number of edges (called maximum separable contact number) of the contact graph of a totally separable packing of $n$ translates of an arbitrary smooth convex body in $\mathbb E^d$ with $d=2,3,4$. In the proofs, linear algebraic and convexity methods are combined with volumetric and packing density estimates based on the underlying isoperimetric (resp., reverse isoperimetric) inequality.

math.MG↗

arXiv2017

Minimizing the mean projections of finite $ρ$-separable packings

A packing of translates of a convex body in the $d$-dimensional Euclidean space $\mathbb{E}^d$ is said to be totally separable if any two packing elements can be separated by a hyperplane of $\mathbb{E}^{d}$ disjoint from the interior of every packing element. We call the packing $\mathcal P$ of translates of a centrally symmetric convex body $\mathbf{C}$ in $\mathbb{E}^d$ a $ρ$-separable packing for given $ρ\geq 1$ if in every ball concentric to a packing element of $\mathcal P$ having radius $ρ$ (measured in the norm generated by $\mathbf{C}$) the corresponding sub-packing of $\mathcal P$ is totally separable. The main result of this paper is the following theorem. Consider the convex hull $\mathbf{Q}$ of $n$ non-overlapping translates of an arbitrary centrally symmetric convex body $\mathbf{C}$ forming a $ρ$-separable packing in $\mathbb{E}^d$ with $n$ being sufficiently large for given $ρ\geq 1$. If $\mathbf{Q}$ has minimal mean $i$-dimensional projection for given $i$ with $1\leq i<d$, then $\mathbf{Q}$ is approximately a $d$-dimensional ball. This extends a theorem of K. Böröczky Jr. [Monatsh. Math. 118 (1994), 41-54] from translative packings to $ρ$-separable translative packings for $ρ\geq 1$.

math.MG↗

arXiv2017

Bounds for totally separable translative packings in the plane

A packing of translates of a convex domain in the Euclidean plane is said to be totally separable if any two packing elements can be separated by a line disjoint from the interior of every packing element. This notion was introduced by G. Fejes Tóth and L. Fejes Tóth (1973) and has attracted significant attention. In this paper we prove an analogue of Oler's inequality for totally separable translative packings of convex domains and then we derive from it some new results. This includes finding the largest density of totally separable translative packings of an arbitrary convex domain and finding the smallest area convex hull of totally separable packings (resp., totally separable soft packings) generated by given number of translates of a convex domain (resp., soft convex domain). Finally, we determine the largest covering ratio (that is, the largest fraction of the plane covered by the soft disks) of an arbitrary totally separable soft disk packing with given soft parameter.

math.MG↗

arXiv2017

The Kneser--Poulsen conjecture for special contractions

The Kneser--Poulsen Conjecture states that if the centers of a family of $N$ unit balls in ${\mathbb E}^d$ is contracted, then the volume of the union (resp., intersection) does not increase (resp., decrease). We consider two types of special contractions. First, a \emph{uniform contraction} is a contraction where all the pairwise distances in the first set of centers are larger than all the pairwise distances in the second set of centers. We obtain that a uniform contraction of the centers does not decrease the volume of the intersection of the balls, provided that $N\geq(1+\sqrt{2})^d$. Our result extends to intrinsic volumes. We prove a similar result concerning the volume of the union. Second, a \emph{strong contraction} is a contraction in each coordinate. We show that the conjecture holds for strong contractions. In fact, the result extends to arbitrary unconditional bodies in the place of balls.

math.MG↗

arXiv2001

Pushing disks apart - The Kneser-Poulsen conjecture in the plane

We give a proof of the planar case of a longstanding conjecture of Kneser (1955) and Poulsen (1954). In fact, we prove more by showing that if a finite set of disks in the plane is rearranged so that the distance between each pair of centers does not decrease, then the area of the union does not decrease, and the area of the intersection does not increase.

math.MG↗