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ON CONTACT NUMBERS OF LOCALLY SEPARABLE UNIT SPHERE PACKINGS
Author(s) -
Bezdek Károly
Publication year - 2021
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/mtk.12102
Subject(s) - mathematics , unit sphere , separable space , combinatorics , ball (mathematics) , circle packing , sphere packing , hyperplane , euclidean space , bounded function , disjoint sets , geometry , mathematical analysis
Abstract The contact number of a packing of finitely many balls in Euclidean d ‐space is the number of touching pairs of balls in the packing. A prominent subfamily of sphere packings is formed by the so‐called totally separable sphere packings: here, a packing of balls in Euclidean d ‐space is called totally separable if any two balls can be separated by a hyperplane such that it is disjoint from the interior of each ball in the packing. Bezdek et al . ( Discrete Math . 339(2) (2016), 668–676) upper bounded the contact numbers of totally separable packings of n unit balls in Euclidean d ‐space in terms of n and d . In this paper, we improve their upper bound and extend that new upper bound to the so‐called locally separable packings of unit balls. We call a packing of unit balls a locally separable packing if each unit ball of the packing together with the unit balls that are tangent to it form a totally separable packing. In the plane, we prove a crystallization result by characterizing all locally separable packings of n unit disks having maximum contact number.