01-10
Successive minima and Lattice points
by Henk, Martin
Preprint series: 01-10, Preprints
- MSC:
- 52C17 Packing and covering in $n$ dimensions, See also {05B40, 11H31}
- 11H31 Lattice packing and covering, See also {05B40, 52C15, 52C17}
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
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