The classical Brunn-Minkowski inequality in the $n$-dimensional Euclidean space asserts that the volume (Lebesgue measure) to the power $1/n$ is a concave functional when dealing with convex bodies (non-empty compact convex sets). It quickly yields, among other results, the classical isoperimetric inequality, which can be summarized by saying that the Euclidean balls minimize the surface area measure (Minkowski content) among those convex bodies with prescribed positive volume. Moreover, it implies the so-called Rogers-Shephard inequality, which provides a sharp upper bound for the volume of the difference set in terms of the volume of the original convex body.
There exist various facets of the previous results, due to their different versions, generalizations, and extensions. In this talk, after recalling the above classical inequalities for the volume, we will discuss and show certain discrete analogues of them for the lattice point enumerator, which gives the number of integer points of a bounded set. Moreover, we will show that these new discrete inequalities imply the corresponding classical results for convex bodies.
This is about joint works with David Alonso-Gutiérrez (Universidad de Zaragoza), David Iglesias (Universidad de Murcia), Eduardo Lucas (Universidad de Murcia), and Artem Zvavitch (Kent State University).