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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1976-0412970-2
PII: S 0002-9939(1976)0412970-2
Keywords: Convex set, covering, density
Article copyright: © Copyright 1976 American Mathematical Society