Interpolating between volume and lattice point enumerator with successive minima

TL;DR

This paper establishes new inequalities linking lattice points, volume, and successive minima of convex bodies, using Blaschke's shaking procedure to derive bounds that unify and extend classical results in the geometry of numbers.

Contribution

It introduces a novel approach to relate lattice point counts and volume via successive minima, utilizing Blaschke's shaking to reduce the problem to anti-blocking bodies.

Findings

Derived an upper bound on lattice points based on successive minima.

Established an inequality connecting volume and successive minima.

Unified classical bounds within a new inequality framework.

Abstract

We study inequalities that simultaneously relate the number of lattice points, the volume and the successive minima of a convex body to one another. One main ingredient in order to establish these relations is Blaschke's shaking procedure, by which the problem can be reduced from arbitrary convex bodies to anti-blocking bodies. As a consequence of our results, we obtain an upper bound on the lattice point enumerator in terms of the successive minima, which is equivalent to Minkowski's upper bound on the volume in terms of the successive minima.