Discrete & Computational Geometry

, Volume 16, Issue 3, pp 305–311

Inequalities for convex bodies and polar reciprocal lattices inRn II: Application ofK-convexity

  • W. Banaszczyk
Article

Abstract

The paper is a supplement to [2]. LetL be a lattice andU ano-symmetric convex body inRn. The Minkowski functionaln ofU, the polar bodyU0, the dual latticeL*, the covering radius μ(L, U), and the successive minima λi,i=1, …,n, are defined in the usual way. Let\(\mathcal{L}_n \) be the family of all lattices inRn. Given a convex bodyU, we define
$$\begin{gathered} mh(U){\text{ }} = {\text{ }}\sup {\text{ }}\max \lambda _i (L,U)\lambda _{n - i + 1} (L^* ,U^0 ), \hfill \\ {\text{ }}L \in \mathcal{L}_n 1 \leqslant i \leqslant n \hfill \\ lh(U){\text{ }} = {\text{ }}\sup {\text{ }}\lambda _1 (L,U) \cdot \mu (L^* ,U^0 ), \hfill \\ {\text{ }}L \in \mathcal{L}_n \hfill \\ \end{gathered} $$
and kh(U) is defined as the smallest positive numbers for which, given arbitrary\(L \in \mathcal{L}_n \) andxRn/(L+U), somey∈L* with ∥yU0sd(xy,Z) can be found. It is proved
$$C_1 n \leqslant jh(U) \leqslant C_2 nK(R_U^n ) \leqslant C_3 n(1 + \log n),$$
, for j=k, l, m, whereC1,C2,C3 are some numerical constants andK(RUn) is theK-convexity constant of the normed space (Rn, ∥∥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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Copyright information

© Springer-Verlag New York Inc 1996

Authors and Affiliations

  • W. Banaszczyk
    • 1
  1. 1.Institute of mathematicsŁódź UniversityŁódźPoland

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