Asymptotic Estimates in Convex Geometry

The proposed research project lies at the intersection of asymptotic geometric analysis and convex geometry. Asymptotic geometric analysis has long adopted a significant part of classical convexity and used many of its methods and techniques. As a consequence, an asymptotic point of view was gradually transferred to convex geometry. The study of geometric problems from a functional analytic and probabilistic point of view enriched classical convexity with a new intuition and proved to be a source of fascinating and challenging geometric problems. Our goal is to contribute to the study of high dimensional convex bodies by blending tools from convexity, integral geometry, and probability, to obtain asymptotically exact versions of geometric and functional inequalities. The main directions of our research are as follows: 1. Asymptotic versions of conjectured classical affine isoperimetric inequalities. We will deal with the systematic analysis of the affine quermassintegrals of high dimensional convex bodies. In particular, we will be interested in the asymptotic version of Lutwaks conjecture which asserts that the affine quermassintegrals of any convex body are isomorphically determined by the corresponding ones of the Euclidean ball. From the applicants previous work with Paouris, we know this conjecture is almost true, up to a logarithm of the dimension factor. Our goal is to bring in new tools from classical convexity and integral geometry to achieve a better estimate, at least for special cases. We also aim to carefully examine the special case of the regular simplex, and the volume distribution of its random projections. The study of this extremal case will help us to extract new important insight into the problem. 2. Correlation inequalities for Gaussian random vectors. A second line of research deals with the correlation inequalities proved by Chen, Paouris and the applicant, that generalize the sharp Young inequality, Nelsons hypercontractivity, their reverse forms, and interpolate between independence and Hölders inequality. Their deep connection with the famous geometric Brascamp-Lieb and Barthes inequalities makes them a strong potential tool in proving sharp geometric inequalities for high dimensional convex bodies. We aim to study the geometry of the eligible exponents in these correlation inequalities, in order to establish new significant results toward a better understanding. A reverse form of the logarithmic Sobolev inequality (by Paouris and the applicant) for log-concave functions is strongly connected to them. We aim to use semigroup techniques, in combination with probabilistic and geometric tools, in order to further investigate this connection and try to remove the log-concavity condition.

During this project we had the opportunity to work at the edge of new developments in the field of our research. The goal was to bring new tools from the convexity machinery into the study of high dimensional convex bodies and vice versa. We aimed at asymptotic versions of classic isoperimetric inequalities, concerning the affine quermassintegrals of a convex body in Rn, and further investigations of certain Brascamp-Lieb type correlation inequalities for Gaussian random vectors and their applications in convex geometry and analysis. Because of the early termination of the project, we have mostly worked on the first part while for the second one we have done some preliminary and preparatory work. We introduced Orlicz mixed affine quermassintegrals, a new generalization of the affine quermassintegrals in Orlicz spaces. We present our work in the paper [1], which has been submitted for publication, where we study these new geometric quantities, proving their invariance under volume preserving linear transformations and their strong ties to certain fundamental results in convex geometry such as the Minkowski and Brunn-Minkowski inequalities. Our integration in the Institute of Discrete Mathematics and Geometry, and especially with Professor Dr. F. Schusters research group, worked ide- ally. We believe that both sides have made significant profits of this project, which provided us an excellent research environment and perfect conditions for our work. 1 Paper, in which we present our work on Orlicz Mixed Affine Quermassintegrals, and it is available at https://arxiv.org/abs/1809.10006.