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2001-10

Successive minima and Lattice points

by Henk, Martin.


Series: 2001-10, Preprints

MSC:
52C17 Packing and covering in $n$ dimensions
11H31 Lattice packing and covering

Abstract:
The main purpose of this note is to prove an upper
bound on the number of lattice points of a centrally symmetric
convex body in terms of the successive minima of the body. This bound
improves on former bounds and narrows the gap towards a lattice
point
analogue of Minkowski's second theorem on successive minima.
Moreover, we present Minkowski's original proof of
his second theorem on successive minima. It turns out that
using the modern mathematical language
his proof is extremely elegant and short,
although it was criticized as lengthy and obscure.

Keywords:
successive minima, convex bodies, lattices, lattice points