Inequalities for the lattice point enumerator

Resumen

The purpose of the thesis project is to establish discrete analogues of known inequalities in the field of Convex Geometry. In this context, discretization is the process of obtaining versions of said inequalities in which the structure and general characteristics are preserved, but where some of the elements (e.g., the measures involved, or the ambient space) are substituted by discrete ones which play a similar role. On the one hand, we will mainly work with lattices, that is, discrete and additive subgroups of the Euclidean space. On the other hand, the usual measure will be the lattice point enumerator (LPE), although the cardinality will also play a relevant role. The methodology employed for this process is similar in all cases. We begin by studying the continuous case, paying special attention to the techniques utilized in the proofs of the inequalities. Next, we study the state of the art in the discrete field, that is, the known results of said field in the lines of research most related to our problem. Finally, we develop discrete techniques that allow us to obtain the desired analogous inequalities in the discrete setting. As usual in Mathematics, the process does not necessarily start with a specifically conjectured theorem and end with its proof, but rather, throughout the delving we determine what does or does not work, until eventually being able to culminate with a result in the desired direction. The results obtained in this project are circumscribed primarily in four distinct problems in Convex Geometry, and as such, they are collected in four respective chapters of the thesis, each usually corresponding to one or more reseach papers where the results have been (or will be) published. The first chapter has its origin in the prior research of the group, including previous thesis projects: the Brunn-Minkowski inequality. Here, we obtain a discrete version via the LPE for linear combinations with arbitrary positive coefficients, thus generalizing prior results. Next, we prove a discrete version for p-combinations, where p is a positive parameter that can either be bigger or smaller than 1 (a point in which a crucial transition of the techniques required happens). Apart from some additional observations, we show that these new inequalities imply their corresponding continuous analogues. In the second chapter we focus on the isoperimetric inequality, perhaps one of the most classical geometric inequalities, and obtain a version for the LPE which implies the original one. Furthermore, we delve deeper into prior studies for the cardinality measure and prove a characterization for the equality case, showing that the extremal sets are precisely the lattice cubes. In the third chapter we switch over to several inequalities by Rogers and Shephard, as well as other related ones. Using diverse techniques, based both on comparing the volumen with the LPE, as well as on adapting the original ones, we obtain multiple discrete analogues (often not comparable to each other) of the original Rogers-Shephard inequality, as well as of their projection-section type inequality. We also prove a discrete analogue of Berwald’s inequality, from which additional versions of the previous inequalities are derived. These new discrete inequalities, as before, allow us to recover the original ones. Finally, in the fourth chapter we study several existing conjectures related to Minkowski’s successive minima, which are functionals of crucial importance in the field of Discrete Geometry. In particular, we obtain powerful comparison results between the volume and LPE functionals via the successive minima, proving both upper and lower bounds that confirm (at least, in order of magnitude) said conjectures, and which moreover provide alternative proofs of several classical inequalities in the field.