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

 

A symmetric star body that tiles but not as a lattice


Author: Sherman K. Stein
Journal: Proc. Amer. Math. Soc. 36 (1972), 543-548
MSC: Primary 52A45
MathSciNet review: 0319058
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Abstract: A classical question in convex bodies runs as follows: ``If translates of a fixed convex body K in Euclidean space can be packed with a certain density, is it possible to find a lattice packing by translates of K that is at least as dense?'' This suggests a similar question for star bodies, which is answered negatively in the present paper. It is shown that there is a centrally-symmetric star body that tiles ten-dimensional Euclidean space but does not tile it in a lattice manner.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1972-0319058-6
PII: S 0002-9939(1972)0319058-6
Keywords: Packing, tiling, convex body, star body, lattices, nonconvex body, abelian group
Article copyright: © Copyright 1972 American Mathematical Society