A short proof of Minkowski's second theorem and its application to some open questions

TL;DR

This paper provides a concise proof of Minkowski's second theorem and applies it to various open problems in convex geometry, including lattice packing density and Ehrhart polynomials.

Contribution

It introduces a notably short proof of Minkowski's second theorem and demonstrates its application to multiple unresolved questions in convex geometry.

Findings

A surprisingly short proof of Minkowski's second theorem

New insights into lattice packing density and convex body anomalies

Applications to Ehrhart polynomial and lattice point enumeration

Abstract

In this paper we present a surprisingly short proof of Minkowski's second theorem. The author hopes there is no mistake in it, though the argument seems to be too plain to contain one. Also, we apply the main construction of the proof to some problems concerning the anomaly of a convex body, the density of the densest lattice packing, lattice point enumerator, and Ehrhart polynomial.