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2000-04

On free planes in lattice ball packings

by Henk, Martin, Ziegler, Günter M., Zong, Chuanming.


Series: 2000-04, Preprints

MSC:
11H31 Lattice packing and covering
52C17 Packing and covering in $n$ dimensions

Abstract:
This note, by studying relations between the length
of shortest lattice vectors and the covering minima of a lattice,
mainly proves that for every $d$-dimensional packing lattice of balls
one can find a $4$-dimensional plane, parallel to a lattice plane,
such that the plane meets none of the balls of the packing, provided the
dimension $d$ is large enough. On the other hand, we show that for
certain ball packing lattices
the highest dimension of such ``free planes'' is far from $d$.

Keywords:
covering minima, homogeneous