Inequalities for convex bodies and polar reciprocal lattices inR
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II: Application ofK-convexity |
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Authors: | W Banaszczyk |
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Institution: | 1. Institute of mathematics, ?ód? University, 90-238, ?ód?, Poland
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Abstract: | The paper is a supplement to 2]. LetL be a lattice andU ano-symmetric convex body inR n . The Minkowski functional? 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 $$\begin{gathered} mh(U){\text{ }} = {\text{ }}\sup {\text{ }}\max \lambda _i (L,U)\lambda _{n - i + 1} (L^* ,U^0 ), \hfill \\ {\text{ }}L \in \mathcal{L}_n 1 \leqslant i \leqslant n \hfill \\ lh(U){\text{ }} = {\text{ }}\sup {\text{ }}\lambda _1 (L,U) \cdot \mu (L^* ,U^0 ), \hfill \\ {\text{ }}L \in \mathcal{L}_n \hfill \\ \end{gathered} $$ and kh(U) is defined as the smallest positive numbers for which, given arbitrary $L \in \mathcal{L}_n $ andx∈R n /(L+U), somey∈L * with ∥y∥ U 0?sd(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, whereC 1,C 2,C 3 are some numerical constants andK(R U n ) 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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