Consequences and Extensions of the Brunn-Minkowski Theorem

In this work we study some extensions and consequences of the fundamental Brunn-Minkowski inequality, using two different approaches: on one hand we deal with the so-called Grünbaum inequality, a beautiful consequence of the Brunn-Minkowski theorem which asserts, roughly speaking, that any hyperplane passing through the centroid divides any compact convex set into two not too small parts; on the other hand we study discrete versions of the Brunn-Minkowski inequality for the lattice point enumerator, that is, the functional counting how many points with integer coordinates are contained in a bounded set.