Local Theory of Banach Spaces and Convex Geometry

In the first part of this research proposal, we want to investigate finite-dimensional subspaces of L1, i.e., the so- called local structure of this space, and related topics, e.g. combinatorial and probabilistic inequalities. To be more precise, we want to study certain classes of finite-dimensional sequence spaces, i.e., generalized Orlicz spaces such as Musielak-Orlicz spaces, Orlicz-Lorentz spaces and Musielak-Orlicz-Lorentz spaces. The goal is to find easily verifiable conditions to decide wether a given Banach space is a subspace of L1 or not. The methods in our approach involve combinatorial and probabilistic inequalities in connection with Orlicz norms. Hence, throughout the proposal, we will take a closer look at those inequalities. In this context, we also want to prove some inversion formulas, telling us, given a certain Orlicz function, how to choose the distribution of the random variables so that the probabilistic expression is equivalent to the given Orlicz norm. Such inversion results immediately provide direct embeddings of certain Orlicz spaces into L1. In recent years, more and more applications of those inequalities appeared, e.g., non-parametric statistics, random matrix theory, convex geometry. In fact, in the third part, we will also take a look at applications in convex geometry. We will study the expectation of support functions, the mean width of random polytopes and their perturbations and the so-called mean outer radii of random polytopes. The probabilistic expressions do appear naturally in this context.

The discipline of asymptotic geometric analysis bridges, in essence, three areas of mathematics, namely, functional analysis, convex geometry, and probability theory. Objects of study are high-dimensional linear structures such as finite-dimensional normed spaces, linear operators on them, high-dimensional convex bodies and the asymptotic behaviour of quantitative parameters as the dimension tends to infinity. The frequency of high-dimensional systems in mathematics itself but also in applied sciences, demands a deep understanding of high-dimensional phenomena. This project contributed to a deepened understanding of the geometric structure of finite-dimensional Banach spaces as the dimension tends to infinity. The main results obtained during this research project relate to characterizations of subspaces of classical Banach spaces with a certain underlying structure, probabilistic methods in Banach spaces theory involving order statistic and the geometry of random convex sets in high-dimensions. Structures of this flavor naturally appear in mathematical physics or theoretical computer science.