Inequalities for convex bodies and polar reciprocal lattices inR
n |
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Authors: | W Banaszczyk |
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Institution: | (1) Institute of Mathematics, Łódź University, 90-238 Łódź, Poland |
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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
p
n
, 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. |
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Keywords: | |
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