On $k^+$-neighbour packings and one-sided Hadwiger configurations

Károly Bezdek and Peter Brass

Abstract: We show that in $d$-dimensional Euclidean space the maximum number of non-overlapping translates of a $d$-dimensional convex body $K$ that can touch $K$ and can lie in a closed supporting half-space of $K$ is always at most $2\cdot 3^{d-1} -1$, with this bound to be reached only if $K$ is an affine image of a $d$-cube. Such one-sided Hadwiger configurations occur at the boundary of finite packings or near the holes in packings of density 0, so this implies that in $d$-dimensional Euclidean space any $k^{+}$-neighbour packing by translates of a $d$-dimensional convex body has positive density for all $k\geq 2\cdot 3^{d-1}$; and there is a $(2\cdot3^{d-1}-1)^+$-neighbour packing by translates of a $d$-cube that has density 0. (A packing is called a $k^+$-neighbour packing if each packing element has at least $k$ neighbours.) This answers an old question of L. Fejes Tóth (1973).

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