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Lattice covering of space with symmetric convex bodies
Author(s) -
Gritzmann Peter
Publication year - 1985
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579300011086
Subject(s) - mathematics , convex body , hyperplane , combinatorics , regular polygon , unit sphere , euclidean space , lattice (music) , upper and lower bounds , affine transformation , geometry , mathematical analysis , convex hull , physics , acoustics
In 1959 C. A. Rogers gave the following estimate for the density ϑ(k) of lattice‐coverings of euclidean d ‐space E d with convex bodies K : ϑ L ( K ) ⩽ dlog 2log e d + c. Here, c is a suitable constant which does not depend on d and K . Moreover, Rogers proved that for the unit ball B d the upper bound can be replaced by c d(log e d )( 1 / 2 )log 2 2 π e, 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 E d , and let k be an integer satisfying k > log 2 log e d + 4 . If there exist at least k hyperplanes H 1 ,…, H k with normals mutually perpendicular and an affine transformation A such that A(K) is symmetrical with respect to H l ,…,H k , respectively, thenϑ L ( K ) ⩽ c d(log e d )1 + log 2 e. 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.