On geometric and functional Grünbaum type inequalities

Résumé

The study of Grünbaum type inequalities has demonstrated to be a very prolific and interesting topic during the last years. Its origin goes back to a classical work by Ascoli from the 30's, lately generalized to higher dimensions by Grünbaum, and published in 1960. It concerns a very natural question involving convex bodies: can one ensure the existence of an ``a priori'' point within the interior of a convex body in such a way that cutting it through this point results in two parts both having a remarkable portion of the total volume? Trying to figure out an answer to the previous question one is led to the centroid (also known as the center of mass) of a convex body. Grünbaum's inequality then asserts that, if K a is convex body with centroid at the origin, then vol(K-)/vol(K) is bounded by (n/(n+1))n . This dissertation approaches three fundamental questions related to Grünbaum’s inequality: • On the one hand, attending to Grünbaum’s original proof, it is natural to wonder about a possible enhanced version of this inequality for the family of those compact convex sets K such that (there exists a hyperplane H for which) the cross-sections volume function is p-concave, with 1/(n-1) < p. On the other hand, one could expect to extend this inequality to compact sets K, not necessarily convex, for which f is p-concave (for some hyperplane H), with p < 1/(n-1). • Attending to the interplay between log-concave and p-concave functions and the geometry of convex sets, it seems reasonable to expect a functional form of Grünbaum's inequality. Moreover, taking into account its connection with the Brunn-Minkowski inequality, one would claim that the Borell-Brascamp-Lieb inequality should play a relevant role in the proof of such an analytic result. • Reading Grünbaum’s inequality from a slightly different perspective, it asserts that for any convex body, there always exists a point within the set (the centroid) in such a way that cutting the body through a hyperplane passing by this point yields two sub-bodies with a substantial proportion of the total volume. This observation prompts a natural inquiry: can we identify a family of points, potentially including the centroid, that share this property? Moreover, are there additional special points that possess similar intriguing characteristics, warranting further investigation? To address these questions, this work starts with an introductory first chapter where we collect some definitions and results that will be needed later on. Thus, in the second chapter we show that by fixing a hyperplane H, one can find a sharp lower bound for the ratio vol(K-)/vol(K) depending on the concavity nature of the cross-sections volume function (parallel to H) of K. Moreover, in the fourth chapter, we go a bit further than the first posed question; specifically, we see that if the cross-sections volume function is ɸ-concave one can bound vol(K-)/vol(K) with a constant depending solely on the function ɸ. Regarding the second question, in the third chapter, we give a simple proof of the functional form of Grünbaum's inequality by using induction on the dimension and the Borell-Brascamp-Lieb inequality. Finally, concerning the last posed question, we present two examples of special points where a Grünbaum-type inequality is not feasible. On the contrary, we show that is possible to extend Grünbaums’s inequality to the case in which the hyperplane H passes by any of the points lying in a whole uniparametric family of r-powered centroids associated with K. The methodology followed in this work has been the usual one in a mathematical project, i.e., it is based on the deep study of previous works in order to develop new techniques related to the posed problems.