Inequalities for convex bodies and polar reciprocal lattices inR n

Abstract

LetL be a lattice and letU be ano-symmetric convex body inR n . The Minkowski functional ∥ ∥ U ofU, the polar bodyU 0, the dual latticeL *, the covering radius μ(L, U), and the successive minima λ i (L,U)i=1,...,n, are defined in the usual way. Let ℒ n be the family of all lattices inR n . Given a pairU,V of convex bodies, we define and kh(U, V) is defined as the smallest positive numbers for which, given arbitraryL∈ℒ n andu∈R n /(L+U), somev∈L * with ∥v∥ V ≤sd(uv, ℤ) can be found. Upper bounds for jh(U, U 0), 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 whenU, V aren-dimensional ellipsoids, rectangular parallelepipeds, or unit balls inl , 1≤p≤∞. The gaps between the upper bounds obtained and the known lower bounds are, roughly speaking, of order at most logn asn→∞. It is also proved that ifU is symmetric through each of the coordinate hyperplanes, then jh(U, U 0) are less thanCn logn for some numerical constantC.