Mahler's work on the geometry of numbers

TL;DR

This paper reviews Mahler's significant contributions to the geometry of numbers, including theorems on lattices, star bodies, and convex bodies, highlighting their impact on the field.

Contribution

It provides a comprehensive overview of Mahler's key results and their influence in the development of the geometry of numbers.

Findings

Mahler's compactness theorem for lattices

Results on star bodies and critical lattices

Estimates for successive minima of convex bodies

Abstract

Mahler has written many papers on the geometry of numbers. Arguably, his most influential achievements in this area are his compactness theorem for lattices, his work on star bodies and their critical lattices, and his estimates for the successive minima of reciprocal convex bodies and compound convex bodies. We give a, by far not complete, overview of Mahler's work on these topics and their impact.