Abstract
The paper is a supplement to [2]. LetL be a lattice andU ano-symmetric convex body inR n. The Minkowski functionalℝ n ofU, the polar bodyU 0, the dual latticeL *, the covering radius μ(L, U), and the successive minima λ i ,i=1, …,n, are defined in the usual way. Let\(\mathcal{L}_n \) be the family of all lattices inR n. Given a convex bodyU, we define
and kh(U) is defined as the smallest positive numbers for which, given arbitrary\(L \in \mathcal{L}_n \) andx∈R n/(L+U), somey∈L * with ∥y∥ U 0⪯sd(xy,Z) can be found. It is proved
, for j=k, l, m, whereC 1,C 2,C 3 are some numerical constants andK(R n U ) is theK-convexity constant of the normed space (R n, ∥∥U). This is an essential strengthening of the bounds obtained in [2]. The bounds for lh(U) are then applied to improve the results of Kannan and Lovász [5] estimating the lattice width of a convex bodyU by the number of lattice points inU.
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This research was supported by KBN Grant 2 P301 019 04.
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Banaszczyk, W. Inequalities for convex bodies and polar reciprocal lattices inR n II: Application ofK-convexity. Discrete Comput Geom 16, 305–311 (1996). https://doi.org/10.1007/BF02711514
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DOI: https://doi.org/10.1007/BF02711514