ON A LEMMA OF MINKOWSKI

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Abstract

In this paper we show that a generalization of a lemma of Minkowski can be applied to solve two problems concerning kissing numbers of convex bodies. In the first application we give a short proof showing that the lattice kissing number of tetrahedra is eighteen. Moreover, it turns out that for any tetrahedron T there exist a unique 18-neighbour lattice packing of T and an essentially unique 16-neighbour lattice packing of T. Secondly we show that for every integer d ≥ 3 there exists a d-dimensional convex body K such that the difference between its translative kissing number and lattice kissing number is at least 2^d-1.

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