Going further in the Lp-Brunn-Minkowski theory a p-difference of convex bodies = ampliando la teoría de Lp de Brunn-Minkowskiuna p-diferencia de cuerpos convexos

Abstract

Resumen Inglés: The main aim of this Doctoral Thesis is to define a new concept, analogous to the Minkowski difference of convex bodies but in the framework of the Lp-Brunn-Minkowski theory, which can be somehow "opposite" to the p-sum of convex bodies, and to study its main properties. Another goal has been to consider the differentiability of the quermassintegrals in the definition parameter of the family of p-parallel bodies. Thus, we start the dissertation establishing the basic notions that will be needed further on. Next, we define the p-difference of two convex bodies in two possible ways -as the largest set that can be p-added to one of them and keeps the result within the other, and in terms of the support functions of the involved convex bodies- and show that, in fact, they are equivalent. We investigate the concavity and the continuity with respect to the Hausdorff metric of this new operation, and define the full system of p-parallel bodies. The third chapter is devoted to study the already mentioned differentiability of the quermassintegrals. In particular, we obtain that the volume is always differentiable in the full range of definition of the parameter, whereas for the remaining quermassintegrals, differentiability only holds for non-negative values of the parameter. In all cases where there is differentiability, we provide the explicit expression for the derivative of the corresponding functional. We conclude the Thesis studying the boundary of the p-inner parallel bodies, in the sense of relating the extreme normal vectors of the original convex body to the ones of its p-inner sets. We also define a new convex body, the so-called p-form body, which will allow us to get bounds for the quermassintegrals of the p-inner parallel bodies, as well as for the derivative, wherever it exists, of the support function with respect to the definition parameter of such a full system of p-parallel sets. The methodology for the attainment of our objectives has been the usual one for basic research in Mathematics: to study papers and books in Convex Geometry and (Lp-) Brunn-Minkowski theory, in order to achieve 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 of new techniques which allow us to solve the outlined problems. In conclusion, we can say that the raised objectives have been achieved. Most of the problems have been successfully solved (feasible definition and main properties of the p-difference in connection with the p-sum; differentiability of the volume and the quermassintegrals; use of p-inner parallel bodies as a tool in order to obtain new inequalities) and, in fact, this can be seen by the three research works which have arisen from this dissertation. Moreover, we think that the content of this Ph.D. Thesis will be the starting point for a deeper development of the Lp-Brunn-Minkowski theory, because of the new tools and techniques that p-inner parallel bodies provide.