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Minkowski's successive minima in convex and discrete geometry

Iskander Aliev, Martin Henk

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.

Minkowski's successive minima in convex and discrete geometry

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 -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.
Paper Structure (6 sections, 107 equations)

This paper contains 6 sections, 107 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Possible Tightenings and Generalizations of Minkowski's theorem
  4. Successive Minima and polarity
  5. Successive Minima and lattice point enumerator
  6. An application: Bombieri-Vaaler extension of Siegel's lemma

Theorems & Definitions (3)

  • proof
  • proof
  • proof