## 00-04

#### On free planes in lattice ball packings

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

**Preprint series:** 00-04, Preprints

**MSC****:**- 11H31 Lattice packing and covering, See also {05B40, 52C15, 52C17}
- 52C17 Packing and covering in $n$ dimensions, See also {05B40, 11H31}

**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 minima, inhomogeneous minimum, ball packings, Korkin-Zolotarev reduced bases

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