Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-1972-0319058-6
MathSciNet review: 0319058
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


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

  • [1] 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)
  • [3] S. K. Stein, Factoring by subsets, Pacific J. Math. 22 (1967), 523-541. MR 36 #2517. MR 0219435 (36:2517)
  • [4] 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)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 52A45

Retrieve articles in all journals with MSC: 52A45


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0319058-6
Keywords: Packing, tiling, convex body, star body, lattices, nonconvex body, abelian group
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society