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Proceedings of the American Mathematical Society

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On a covering property of convex sets


Author: H. Groemer
Journal: Proc. Amer. Math. Soc. 59 (1976), 346-352
MSC: Primary 52A45
MathSciNet review: 0412970
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Abstract: Let $ \{ {K_1},\;{K_2}, \ldots \} $ be a class of compact convex subsets of euclidean $ n$-space with the property that the set of their diameters is bounded. It is shown that the sets $ {K_i}$ can be rearranged by the application of rigid motions so as to cover the total space if and only if the sum of the volumes of all the sets $ {K_i}$ is infinite. Also, some statements regarding the densities of such coverings are proved.


References [Enhancements On Off] (What's this?)

  • [1] Gulbank D. Chakerian, Covering space with convex bodies, The geometry of metric and linear spaces (Proc. Conf., Michigan State Univ., East Lansing, Mich., 1974) Springer, Berlin, 1975, pp. 187–193. Lecture Notes in Math., Vol. 490. MR 0407722
  • [2] G. D. Chakerian and H. Groemer, On classes of convex sets that permit plane coverings, Israel J. Math. 19 (1974), 305–312. MR 0367818
  • [3] Helmut Groemer, Existenzsätze für Lagerungen im Euklidischen Raum, Math. Z. 81 (1963), 260–278 (German). MR 0163222
  • [4] H. Hadwiger, Volumschätzung für die einen Eikörper überdeckenden und unterdeckenden Parallelotope, Elem. Math. 10 (1955), 122–124 (German). MR 0077154
  • [5] A. Kosiński, A proof of an Auerbach-Banach-Mazur-Ulam theorem on convex bodies, Colloq. Math. 4 (1957), 216–218. MR 0086324

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0412970-2
Keywords: Convex set, covering, density
Article copyright: © Copyright 1976 American Mathematical Society