Lattices and Minkowski’s Theorem

Abstract

This chapter is a quick excursion into the geometry of numbers, a field where number-theoretic results are proved by geometric arguments, often using properties of convex bodies in R^d. We formulate the simple but beautiful theorem of Minkowski on the existence of a nonzero lattice point in every symmetric convex body of sufficiently large volume. We derive several consequences, concluding with a geometric proof of the famous theorem of Lagrange claiming that every natural number can be written as the sum of at most 4 squares.