A symmetric star body that tiles but not as a lattice

Stein, Sherman K.

doi:10.1090/S0002-9939-1972-0319058-6

A symmetric star body that tiles but not as a lattice
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by Sherman K. Stein PDF

Proc. Amer. Math. Soc. 36 (1972), 543-548 Request permission

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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