Asymptotic Shape of Finite Packings

Jr., KáRoly Böröczky ; Schnell, Uwe

Canadian journal of mathematics, Tome 50 (1998) no. 1, pp. 16-28

Résumé

Let $K$ be a convex body in ${{\mathbf{E}}^{d}}$ and denote by ${{C}_{n}}$ the set of centroids of $n$ non-overlapping translates of $K$ . For $\varrho \,>\,0$ , assume that the parallel body conv ${{C}_{n}}\,+\,\varrho K$ of conv ${{C}_{n}}$ has minimal volume. The notion of parametric density (see [21]) provides a bridge between finite and infinite packings (see [4] or [14]). It is known that there exists a maximal ${{\varrho }_{s}}(K)\,\ge \,1/(32{{d}^{2}})$ such that conv ${{C}_{n}}$ is a segment for $\varrho \,<\,{{\varrho }_{s}}$ (see [5]). We prove the existence of a minimal ${{\varrho }_{c}}(K)\,\le \,d\,+\,1$ such that if $\varrho \,>\,{{\varrho }_{c}}$ and $n$ is large then the shape of conv ${{C}_{n}}$ can not be too far from the shape of $K$ . For $d\,=\,2$ , we verify that ${{\varrho }_{s\,}}\,=\,{{\varrho }_{c}}$ . For $d\,\ge \,3$ , 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.

Détail

DOI : 10.4153/CJM-1998-002-5

Mots-clés : 52C17, 05B40

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