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Lattice covering of space with symmetric convex bodies

Published online by Cambridge University Press:  26 February 2010

Peter Gritzmann
Peter Gritzmann
Affiliation:
Math. Inst. Univ. Siegen, Holderlinstrasse 3, D-5900 Siegen, Federal Republic of Germany.
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Abstract

In 1959 C. A. Rogers gave the following estimate for the density ϑ(k) of lattice-coverings of euclidean d-space Ed with convex bodies . Here, c is a suitable constant which does not depend on d and K. Moreover, Rogers proved that for the unit ball Bd the upper bound can be replaced by , which is, of course a major improvement. In the present paper we show that such an improvement can be obtained for a larger class of convex bodies. In particular, we prove the following theorem. Let K be a convex body in Ed, and let k be an integer satisfying k > log2 loge d + 4. If there exist at least k hyperplanes H1,…, Hk with normals mutually perpendicular and an affine transformation A such that A(K) is symmetrical with respect to Hl,…,Hk, respectively, then . Actually, for a bound of this type we do not even need any symmetry assumption. In fact, some weaker properties concerning shadow boundaries will suffice.

Type
Research Article
Information
Mathematika , Volume 32 , Issue 2 , December 1985 , pp. 311 - 315
Copyright
Copyright © University College London 1985

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References

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