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Search: MSC category 05B40 ( Packing and covering [See also 11H31, 52C15, 52C17] )

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1. CJM 1998 (vol 50 pp. 16)

Böröczky, Károly; Schnell, Uwe
Asymptotic shape of finite packings
Let $K$ be a convex body in $\ed$ and denote by $\cn$ the set of centroids of $n$ non-overlapping translates of $K$. For $\varrho>0$, assume that the parallel body $\cocn+\varrho K$ of $\cocn$ has minimal volume. The notion of parametric density (see~\cite{Wil93}) provides a bridge between finite and infinite packings (see~\cite{BHW94} or~\cite{Hen}). It is known that there exists a maximal $\varrho_s(K)\geq 1/(32d^2)$ such that $\cocn$ is a segment for $\varrho<\varrho_s$ (see~\cite{BHW95}). We prove the existence of a minimal $\varrho_c(K)\leq d+1$ such that if $\varrho>\varrho_c$ and $n$ is large then the shape of $\cocn$ can not be too far from the shape of $K$. For $d=2$, we verify that $\varrho_s=\varrho_c$. For $d\geq 3$, we present the first example of a convex body with known $\varrho_s$ and $\varrho_c$; namely, we have $\varrho_s=\varrho_c=1$ for the parallelotope.

Categories:52C17, 05B40

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