On roots of general Steiner type polynomial

Abstract

The main aim of this Doctoral Thesis is to investigate the structure and behavior of the set of roots of those n-th degree polynomials whose coefficients (up to the combinatorial numbers) form a log-convex sequence, among which we find the dual Steiner polynomials, and to study the Blaschke problem in the setting of the so-called dual Brunn-Minkowski theory. We start the dissertation with a first chapter where the concepts and results that will be needed later are collected. We explain in detail the Blaschke problem and its relation with the type of roots of the Steiner polynomial. We also dedicate a section to studying the so-called dual Brunn-Minkowski theory, and finally we collect some known results and properties on real polynomials. In the second chapter we study the behaviour of the roots of the Steiner polynomial when we consider 2- and 3-dimensional convex bodies, embedded in a higher dimensional Euclidean space. In this case, we prove that the set of 2-dimensional convex bodies embedded in an n-dimensional Euclidean space whose Steiner polynomial has only real roots, contains the corresponding one in dimension n+1. However, in the 3-dimensional case, we find a counterexample that shows us that this inclusion is false, and that it is verified when we make the corresponding jumping into dimension n+2. In the third chapter we study the behavior of what we will call log-convex coefficients polynomials, this is, those ones whose coefficients (up to the combinatorial numbers) satisfy a_i^2<=a_{i-1}a_{i+1}. We consider the set of roots of these polynomials (here we differentiate that the coefficients are all positive or that some of them may be zero) contained in the upper halfplane, and we prove that both sets are convex cones, among other properties. The main result provides us with the precise description of these cones for any dimension n>=3. In the last chapter we investigate the corresponding Blaschke problem in the dual setting. We characterize the dual Blaschke diagram and prove that it is simply connected and not closed. This allows us to obtain a new characterization of the dual quermassintegrals in dimension n=3 by means of the dual Aleksandrov-Fenchel inequalities, and in this way we can determine the cone of roots of the dual Steiner polynomials for n=3. We also get bounds for the moduli and real and imaginary parts of the roots of dual Steiner polynomials in terms of the inner and outer radii. In the last section we prove that the dual Steiner polynomial with weights associated to a probability measure on the positive real line admits an integral representation and, furthermore, we show that the classical dual Steiner functional can be obtained as one of these generalized functionals for a particular "limit" measure. 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 Convexity and (dual) Theory of Brunn-Minkowski, in order to get the necessary background to address the posed questions; the study of the previously known results, in order to establish the starting points in our research; the development of new techniques which allow us to solve the outlined problems; and the presentation of the results in conferences and research meetings, as well as their publication in impact journals with recognized international prestige. In conclusion, we can say that the raised objectives have been achieved. The problems have been successfully solved (we generalize the dual Steiner polynomial and obtain properties of their roots), which can be seen by the research papers and the different communications in conferences that have arisen from this dissertation.