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