On the Illumination of Three Dimensional Convex Bodies with Affine Plane Symmetry

Abstract

In 1960, Hadwiger and Boltyanski independently posed equivalent versions of the same question: is it possible to illuminate any n-dimensional convex body by 2n light sources? The affirmative answer to this question is called the Boltyanski-Hadwiger Illumination Conjecture. It is one of the best known open problems in Discrete Geometry and derives some of this prominence from its close relationship to the highly studied art gallery problems and from its equivalence to the Levi-Gohberg-Markus Covering Conjecture. In the last fifty-five years, many partial results have been proved. For example, Dekster proved that eight directions illuminate three-dimensional convex bodies with affine plane symmetry. The central feature of this thesis is a rigorous exposition of most cases from Dekster's proof. Three non-trivial theorems play a significant role in the proof: the John-Löwner Theorem, the Blaschke Selection Theorem and Mazur’s Finite Dimensional Density Theorem. Their proofs form another important part of this work.