The Set of Packing and Covering Densities of Convex Disks

Discrete & Computational Geometry, 1998

We prove that for every convex disk C in the plane there exists a double-lattice covering of the plane with copies of C with density 0 < 1.2281772. This improves the best previously known upper bound 0 < 8/(3 + 24'-3) ~ 1.2376043, due to Kuperberg, but it is still far from the conjectured value t9 = 27r/3~,'3 ~ 1.2091993.

Lecture Notes in Computer Science, 2021

We consider the planar two-center problem for a convex polygon: given a convex polygon in the plane, find two congruent disks of minimum radius whose union contains the polygon. We present an O(n log n)-time algorithm for the two-center problem for a convex polygon, where n is the number of vertices of the polygon. This improves upon the previous best algorithm for the problem.

2010

One of the basic problems in discrete geometry is to determine the most efficient packing of congruent replicas of a given convex set K in the plane or in space. The most commonly used measure of efficiency is density. Several types of the problem arise depending on the type of isometries allowed for the packing: packing by translates, lattice packing, translates and point reflections, or all isometries. Due to its connections with number theory, crystallography, etc., lattice packing has been studied most extensively. In two dimensions the theory is fairly well developed, and there are several significant results on lattice packing in three dimensions as well. This article surveys the known results, focusing on the most recent progress. Also, many new problems are stated, indicating directions in which future development of the general packing theory in three dimensions seems feasible.

Discrete & Computational Geometry, 2016

In [BL] in relation to the unsolved Bang's plank problem (1951) we obtained a lower bound for the sum of relevant measures of cylinders covering a given d-dimensional convex body. In this paper we provide the packing counterpart of these estimates. We also extend bounds to the case of r-fold covering and packing and show a packing analog of Falconer's results ([Fa]).

SIAM Journal on Computing, 2011

Let P be a simple polygon, and let Q be a set of points in P . We present an almost-linear time algorithm for computing a minimum cover of Q by disks that are contained in P . We generalize the algorithm above, so that it can compute a minimum cover of Q by homothets of a fixed compact convex set of constant description complexity O that are contained in P . This improves previous results of Katz and Morgenstern . We also consider the disk-cover problem when Q is contained in a (not too wide) annulus, and present an O(|Q| log |Q|) algorithm for this case.

Monatshefte für Mathematik, 2011

A simple n-gon is a polygon with n edges with each vertex belonging to exactly two edges and every other point belonging to at most one edge. Brass [2] asked the following question: For n ≥ 5 odd, what is the maximum perimeter of a simple n-gon contained in a Euclidean unit disk? In 2009, Audet, Hansen and Messine [1] answered this question, and showed that the optimal configuration is an isosceles triangle with a multiple edge, inscribed in the disk. In this note we give a shorter and simpler proof of their result, which we generalize also for hyperbolic disks, and for spherical disks of sufficiently small radii.

Discrete & Computational Geometry, 1992

Let B be a compact convex body symmetric around 0 in R 2 which has nonempty interior, i.e., the unit ball of a two-dimensional Minkowski space. The self-packing radius p(m, B) is the smallest t such that tB can be packed with m translates of the interior of B. For m < 6 we show that the self-packing radius p(m, B) = 1 + 2#t(m, B) where ~t(m, B) is the Minkowski length of the side of the largest equilateral m-gon inscribed in B (measured in the Minkowski metric determined by B). We show p(6, B) = p(7, B) = 3 for all B, and determine most of the largest and smallest values of p(m, B) for m < 7. For all m we have 6(B)/-2 < p(m, B) < \~j/ + 1, where 3(B) is the packing density of B in •2

Motivated by biological questions, we study configurations of equal-sized disks in the Euclidean plane that neither pack nor cover. Measuring the quality by the probability that a random point lies in exactly one disk, we show that the regular hexagonal grid gives the maximum among lattice configurations.

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.

arXiv (Cornell University), 2023

A classical result of Dowker (Bull. Amer. Math. Soc. 50: 120-122, 1944) states that for any plane convex body K in the Euclidean plane, the areas of the maximum (resp. minimum) area convex n-gons inscribed (resp. circumscribed) in K is a concave (resp. convex) sequence. It is known that this theorem remains true if we replace area by perimeter, the Euclidean plane by an arbitrary normed plane, or convex n-gons by disk-n-gons, obtained as the intersection of n closed Euclidean unit disks. The aim of our paper is to investigate these problems for C-n-gons, defined as intersections of n translates of the unit disk C of a normed plane. In particular, we show that Dowker's theorem remains true for the areas and the perimeters of circumscribed C-n-gons, and the perimeters of inscribed C-n-gons. We also show that in the family of origin-symmetric plane convex bodies, for a typical element C with respect to Hausdorff distance, Dowker's theorem for the areas of inscribed C-n-gons fails.