Ih Application of K-Convexity*

The paper is a supplement to [2]. Let L be a lattice and U an o-symmetric convex body inR n. The Minkowski functional I[ I1~ of U, the polar body U ~ the dual lattice L*, the covering radius/z(L, U), and the successive minima ~.i(L, U), i = 1 . . . . . n, are defined in the usual way. Let 12, be the family of all lattices in R". Given a convex body U, we define mh(U) = sup max ~.i(L, U)~.n-i+l(L*, U~ L~s l< i<n lh(U) = sup Zl(L, U) 9 U~ L E L n and kh(U) is defined as the smallest positive number s for which, given arbitrary L 6 12, and x ~ R"\(L + U), some y 6 L* with IlYlluo < sd(xy, Z) can be found. It is proved that Cln < jh(U) < C2nK(R~) < C3n(1 + logn), for j = k, 1, m, where C1, C2, C3 are some numerical constants and K(R"v) is the Kconvexity constant of the normed space (R", II IIu). 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 Lov~sz [5] estimating the lattice width of a convex body U by the number of lattice points in U. This paper is a supplement to the earlier paper [2]. We recall briefly the notation introduced there. By/2n and Cn we denote, respectively, the family of all n-dimensional lattices and the family of all symmetric convex bodies in R n. Let L ~ /~n and U ~ Cn. By L* and U ~ we denote, respectively, the dual lattice and the polar body, defined in * This research was supported by KBN Grant 2 P301 019 04.