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On Free Planes in Lattice Ball Packings
Author(s) -
Henk Martin,
Ziegler Günter M.,
Zong Chuanming
Publication year - 2002
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609301008888
Subject(s) - mathematics , lattice (music) , ball (mathematics) , lattice plane , maxima and minima , combinatorics , sphere packing , geometry , plane (geometry) , condensed matter physics , reciprocal lattice , mathematical analysis , physics , optics , acoustics , diffraction
This note, by studying the relations between the length of the shortest lattice vectors and the covering minima of a lattice, proves that for every d ‐dimensional packing lattice of balls one can find a four‐dimensional plane, parallel to a lattice plane, such that the plane meets none of the balls of the packing, provided that the dimension d is large enough. Nevertheless, for certain ball packing lattices, the highest dimension of such ‘free planes’ is far from d .