MINKOWSKI’S SUCCESSIVE MINIMA AND APPLICATIONS TO NUMBER THEORY

Abstract

In this thesis, we review a number of crucial objects and tools from classical convex geometry. We pay special attention to investigate the conditions under which convex shapes in Euclidean space contain lattice points. We begin by introducing the fundamental concepts of convexity, followed by an exploration of the geometry of numbers and lattice theory. We then examine two central theorems, Pick’s Theorem and Minkowski’s Theorem, which provide insights into the existence and distribution of lattice points within convex bodies, as well as their relationship to the volume of these bodies. Finally, we discuss several applications related to number theory.