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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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- G. Losey, Note on a theorem of Zaremba, J. Combinatorial Theory 6 (1969), 208-209. MR 39 #1333. MR 0239979 (39:1333)
- [2]
- C. A. Rogers, Packing and covering, Cambridge Tracts in Math. and Math. Phys., no. 54, Cambridge Univ. Press, New York, 1964. MR 30 #2405. MR 0172183 (30:2405)
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- S. K. Stein, Factoring by subsets, Pacific J. Math. 22 (1967), 523-541. MR 36 #2517. MR 0219435 (36:2517)
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- M. R. Von Wolff, A star domain with densest admissible point set not a lattice, Acta Math. 108 (1962), 53-60. MR 26 #2400. MR 0144859 (26:2400)
- [5]
- S. K. Zaremba, Covering problems concerning Abelian groups, J. London Math. Soc. 27 (1952), 242-246. MR 13, 817. MR 0047036 (13:817g)
- [6]
- H. Zassenhaus, Modern developments in the geometry of numbers, Bull. Amer. Math. Soc. 67 (1961), 427-439. MR 24 #A1887. MR 0132040 (24:A1887)
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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
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