Inequalities for convex bodies and polar reciprocal lattices in Rn

Abstract

Let L be a lattice and let U be an o-symmetric convex body in Rn . The Minkowski functional ∥ ∥U of U, the polar body U0, the dual lattice L*, the covering radius μ(L, U), and the successive minima λi (L,U)i=1,..., n, are defined in the usual way. Let ℒn be the family of all lattices in Rn . Given a pair U, V of convex bodies, we define[Figure not available: see fulltext.] and kh(U, V) is defined as the smallest positive number s for which, given arbitrary L∈ℒn and u∈Rn /(L+U), some v∈L* with ∥v∥V ≤sd(uv, ℤ) can be found. Upper bounds for jh(U, U0), j=k, l, m, belong to the so-called transference theorems in the geometry of numbers. The technique of Gaussian-like measures on lattices, developed in an earlier paper [4] for euclidean balls, is applied to obtain upper bounds for jh(U, V) in the case when U, V are n-dimensional ellipsoids, rectangular parallelepipeds, or unit balls in lpn, 1≤p≤∞. The gaps between the upper bounds obtained and the known lower bounds are, roughly speaking, of order at most log n as n→∞. It is also proved that if U is symmetric through each of the coordinate hyperplanes, then jh(U, U0) are less than Cn log n for some numerical constant C. © 1995 Springer-Verlag New York Inc.