A discrete analogue for Minkowski's second theorem on successive minima

TL;DR

This paper presents a discrete analogue of Minkowski's second theorem on successive minima for 3D convex bodies, proposing a new inequality involving lattice points and a conjecture for higher dimensions.

Contribution

It introduces a discrete version of Minkowski's theorem for 3D convex bodies and proposes a stronger conjecture for extending the result to higher dimensions.

Findings

Established an inequality relating lattice points and successive minima in 3D

Proposed a stronger conjecture for higher-dimensional generalization

Suggested a proof strategy via induction on dimension

Abstract

The main result of this paper is an inequality relating the lattice point enumerator of a 3-dimensional, 0-symmetric convex body and its successive minima. This is an example of generalization of Minkowski's theorems on successive minima, where the volume is replaced by the discrete analogue, the lattice point enumerator. This problem is still open in higher dimensions, however, we introduce a stronger conjecture that shows a possibility of proof by induction on the dimension.