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On $$k^+$$-neighbour packings and one-sided Hadwiger configurations. (English) Zbl 1041.52010
Let $$K$$ be a convex body with nonempty interior in Euclidean $$d$$-space.
The authors show that the maximum number of non-overlapping translates of $$K$$ that can touch $$K$$ and can lie in a closed supporting half-space of that body is at most $$2\cdot 3^{d-1}-1$$, with equality only for $$K$$ an affine image of a $$d$$-cube. Such configurations are interesting since they occur for example in the “boundary regions” of finite packings. A packing is called a $$k^+$$-neighbour packing if any packing element has at least $$k$$ neighbours. From the result above it follows then that in Euclidean $$d$$-space any $$k^+$$-neighbour packing by translates of a non-degenerate convex body has positive density for all $$k$$ not smaller than $$2\cdot 3^{d-1}$$, and that there is a $$(2\cdot 3^{d-1})^+$$-neighbour packing by translates of a $$d$$-cube having density 0.

##### MSC:
 52C17 Packing and covering in $$n$$ dimensions (aspects of discrete geometry)
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