Successive Minima and Lattice Points

TL;DR

This paper establishes a tighter upper bound on lattice points within centrally symmetric convex bodies using successive minima, and provides a concise proof of Minkowski's second theorem based on original ideas.

Contribution

It introduces an improved bound on lattice points related to successive minima and offers a shorter, clearer proof of Minkowski's second theorem.

Findings

Enhanced upper bound on lattice points in convex bodies

A shorter, more transparent proof of Minkowski's second theorem

Progress towards a lattice point analogue of Minkowski's second theorem

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. Minkowski's proof of his second theorem is rather lengthy and it was also criticised as obscure. We present a short proof of Minkowski's second theorem on successive minima, which, however, is based on the ideas of Minkowski's proof.