Naszódi, Márton and Salcher-Konrad, Maximilian 
ORCID: 0000-0002-5628-5266
(2022)
Contacts in totally separable packings in the place and in high dimensions.
Journal of Computational Geometry.
ISSN 1920-180X
(In Press)
Abstract
We study the contact structure of totally separable packings of translates of a convex body K in Rd, that is, packings where any two translates of the packing have a separating hyperplane that does not intersect the interior of any translate in the packing. The separable Hadwiger number Hsep(K) of K is defined to be the maximum number of translates touched by a single translate, with the maximum taken over all totally separable packings of translates of K. We show that for each d 8, there exists a smooth and strictly convex K in Rd with Hsep(K) > 2d, and asymptotically, Hsep(K) = (3= p 8)d . We show that Alon’s packing of Euclidean unit balls such that each translate touches at least 2 p d others whenever d is a power of 4, can be adapted to give a totally separable packing of translates of the `1-unit ball with the same touching property. We also consider the maximum number of touching pairs in a totally separable packing of n translates of any planar convex body K. We prove that the maximum equals b2n 2 p nc if and only if K is a quasi hexagon, thus completing the determination of this value for all planar convex bodies.
| Item Type: | Article |
|---|---|
| Additional Information: | © The Author(s). |
| Divisions: | Personal Social Services Research Unit |
| Date Deposited: | 31 Oct 2022 10:24 |
| Last Modified: | 15 Sep 2023 17:23 |
| URI: | http://eprints.lse.ac.uk/id/eprint/117211 |
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