Minkowski's successive minima in convex and discrete geometry

TL;DR

This survey explores Minkowski's successive minima and their significant influence on convex and discrete geometry, highlighting their relations to various functionals and applications like Siegel's lemma.

Contribution

It reviews the impact of successive minima on convex geometry, discusses their connections to other functionals, and presents new conjectures and applications.

Findings

Relations between successive minima and lattice point enumeration

Connections to intrinsic volumes and other functionals

Application to a version of Siegel's lemma

Abstract

In this short survey we want to present some of the impact of Minkowski's successive minima within Convex and Discrete Geometry. Originally related to the volume of an $o$-symmetric convex body, we point out relations of the successive minima to other functionals, as e.g., the lattice point enumerator or the intrinsic volumes and we present some old and new conjectures about them. Additionally, we discuss an application of successive minima to a version of Siegel's lemma.