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

Abstract

The paper is a supplement to [2]. LetL be a lattice andU ano-symmetric convex body inRn. The Minkowski functionalźn ofU, the polar bodyU0, the dual latticeL*, the covering radius μ(L, U), and the successive minima źi,i=1, ź,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 $$L \in \mathcal{L}_n $$ andxźRn/(L+U), someyźL* with źyźU0źsd(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, źźU). 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 Lovasz [5] estimating the lattice width of a convex bodyU by the number of lattice points inU.