On discrete Brunn-Minkowski type inequalities

Abstract

The famous Brunn-Minkowski inequality establishes that the n-th root of the volume of compact (and convex) sets is a concave function. The main aim of this Doctoral Thesis is to study Brunn-Minkowski type inequalities in the discrete setting, i.e., replacing compact sets by finite sets of points, and the volume by the cardinality. We start the dissertation establishing the basic notions that will be needed further on. Next, in this first chapter, we study the Brunn-Minkowski inequality, and we verify that it is not possible to get an "immediate translation" of it for finite sets. So, it will be needed either to change the structure of the inequality, or to modify one of the involved sets. In the last part of this chapter we analyze known discrete Brunn-Minkowski type inequalities having a different structure, specially emphasizing the Gardner & Gronchi inequality, for which we propose a new method to efficiently compute the bound. The second chapter is devoted to determining discrete versions of the Brunn-Minkowski inequality, keeping its classical form, and so, modifying one of the involved sets. In order to do it, we consider two different constructions: in the first one we add points to one of the sets in a recursive way, whereas in the second one particular points are removed. Using this constructions we get two new discrete (equivalent) versions of the Brunn-Minkowski inequality, proving also that they imply the classical Brunn-Minkowski inequality for the volume of compact sets. We conclude the chapter by providing upper and lower bounds for the cardinality of the new sets, which shows that the number of added/removed points in these constructions is, somehow, controlled. Finally, in the third chapter we use the previous techniques in order to obtain discrete versions of the so-called Borell-Brascamp-Lieb inequality, a functional result which generalizes the Brunn-Minkowski inequality. There exist two equivalent versions of the classical result of Borell-Brascamp-Lieb, which we introduce in the first part of the chapter. In both cases we show the corresponding discrete version, but using two different ways of measuring: the cardinality for the first one, and the lattice point enumerator in the second one. We prove that both can be also used to deduce the corresponding classical Borell-Brascamp-Lieb inequalities. The methodology for the achievement of our objectives has been the usual one for basic research in Mathematics: the in-depth study of papers and books in Convex and Discrete Geometry, in order to attain the necessary background to address the posed questions, a detailed analysis of the previously known results, in order to establish the starting points in our research, and the development and creation of new techniques which allow us to solve the outlined problems. In conclusion, we can say that the raised objectives have been achieved. The problems have been successfully solved (to obtain new discrete Brunn-Minkowski and Borell-Brascamp-Lieb type inequalities) and, in fact, this can be seen by the four research papers and the different communications in conferences which have arisen from this dissertation. We also believe that the content of this Ph.D. Thesis will allow further research on discrete inequalities, because of the new constructions and techniques that have been developed