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). This result has become not only a cornerstone of the Brunn-Minkowski theory, but also a powerful tool in other related fields of mathematics.
In this talk we will make a brief walk on this inequality, as well as on its extensions to the Lp-setting, for non-negative values of p. Then, we will move to the discrete world, either considering the integer lattice endowed with the cardinality, or working with the lattice point enumerator, which provides with the number of integer points contained in a given convex body: we will discuss and show certain discrete analogues of the above mentioned Brunn-Minkowski type inequalities in both cases.
This is about joint works with Eduardo Lucas and Jesús Yepes Nicolás.