On contact graphs of totally separable domains

Abstract

Contact graphs have emerged as an important tool in the study of translative packings of convex bodies. The contact number of a packing of translates of a convex body is the number of edges in the contact graph of the packing, while the Hadwiger number of a convex body is the maximum vertex degree over all such contact graphs. In this paper, we investigate the Hadwiger and contact numbers of totally separable packings of convex domains, which we refer to as the separable Hadwiger number and the separable contact number, respectively. We show that the separable Hadwiger number of any smooth convex domain is 4 and the maximum separable contact number of any packing of n translates of a smooth strictly convex domain is 2n − 2 √ n . Our proofs employ a characterization of total separability in terms of hemispherical caps on the boundary of a convex body, Auerbach bases of finite dimensional real normed spaces, angle measures in real normed planes, minimal perimeter polyominoes and an approximation of smooth o-symmetric strictly convex domains by, what we call, Auerbach domains.for every x ∈ R d . We denote the corresponding normed space by (R d , · K ). If K = B d , then we denote the norm · B d simply by · . Thus E d = (R d , · ). A d-dimensional convex body K is said to be smooth if at every point on the boundary bd K of K, the body K is supported by a unique hyperplane of E d and strictly convex if the boundary of K contains no nontrivial line segment. Given d-dimensional convex bodies K and L, their Minkowski sum (vector sum) is denoted by K + L and defined aswhich is a convex body in E d .The kissing number problem asks for the maximum number k(d) of non-overlapping translates of B d that can touch B d in E d . Clearly, k(2) = 6. However, the value of k(3) remained a mystery for several hundred years and caused a famous argument between Newton and Gregory in the 17th century. To date, the only *

On contact graphs of totally separable packings in low dimensions

“…In this paper we investigate the maximum vertex degree (called separable Hadwiger number ), as well as the maximum number of edges (called the maximum separable contact number ) of the contact graphs of totally separable packings by a given number of translates of a smooth or strictly convex body K in E d . This extends and generalizes the results of [4] and [6]. The details follow.…”

“…We note that part (a) of Theorem 1 was proved for d = 2 and smooth o-symmetric convex domains in [4]. We prove Theorem 1 in Section 3.…”

“…In [4] it is proved that c sep (K, n) = ⌊2n − 2 √ n⌋ holds for any o-symmetric smooth strictly convex domain K and any n > 1. Thus, one may wonder whether the same statement holds for any smooth o-symmetric convex domain K. We prove Theorem 3 in Section 4.…”

“…Using the same method, it is easy to see that Minkowski's above statement applies to totally separable packings as well. (See also [4].) Thus, from this point on, we only consider o-symmetric convex bodies.…”

From the separable Tammes problem to extremal distributions of great circles in the unit sphere

“…Finally, we conclude our investigations with some higher dimensional analogue statements. We note that totally separable packings and coverings have been already studied in Euclidean (resp., normed) spaces ( [2], [3], [4], [5], [6], [7], [8], [9], [16], [17], [26]), but not yet in spherical spaces, which is the main target of this paper. The details are as follows.…”