A Polyhedral Perspective of the Lonely Runner Conjecture

Abstract

The main focus of this thesis is the ‘Lonely Runner Conjecture’, an open problem that has remained unsolved for over half a century. The problem comes in different flavours. As a result, solving the Conjecture provides us with new information in various fields of Mathematics. First, we take a tour of Polyhedral theory and Discrete Geometry. On this tour, we will have a peek into concepts like ‘Polyhedra’, ‘Ehrhart theory’ and ‘Lattices’, and the field of ‘Geometry of Numbers’. Then, we go over the well-known results about the Conjecture. While doing so, we shall see a detailed description of the ‘Lonely Runner polyhedron’, and the results obtained using it. Finally, we make use of the various concepts that we learnt and obtain a few new results.