99-30
Segments in ball packings
by Henk, Martin; Zong, Chuanming
Preprint series: 99-30, Preprints
Abstract: Denote by $B^n$ the $n$-dimensional unit ball centred at ${\bf o}$. It is known that in every lattice packing of $B^n$ there is a cylindrical hole of infinite length whenever $n\ge 3$. As a counterpart, this note mainly proves the following result: {\it For any fixed $\epsilon $, $\epsilon > 0$, there exist a periodic point set $P(n, \epsilon )$ and a constant $c(n, \epsilon )$ such that $B^n+P(n, \epsilon )$ is a packing in $R^n$, and the length of the longest segment contained in $R^n\setminus \{ {\rm int} (\epsilon B^n)+P(n, \epsilon )\}$ is bounded by $c(n, \epsilon )$ from above.} Generalizations and applications are presented.}
Keywords: ball packings, periodic packings
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