### 1999-30

#### Segments in ball packings

Series: 1999-30, Preprints

MSC:
52C17 Packing and covering in $n$ dimensions

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