Analytic and Probabilistic Problems in Discrete Geometry

TL;DR

This thesis explores two discrete geometry problems: the polarization problem in Hilbert spaces using analytic and combinatorial methods, and the probabilistic behavior of convex chains in random triangles, revealing asymptotic and concentration results.

Contribution

It provides new solutions to the strong polarization problem and establishes asymptotic behavior and concentration results for convex chains in probabilistic geometry.

Findings

The orthonormal system is the only full-dimensional locally extremal system.

Expected length of the longest convex chain scales as n^{1/3} with a constant between 1.5 and 3.5.

Strong concentration results imply a limit shape for the longest convex chains.

Abstract

The thesis concentrates on two problems in discrete geometry, whose solutions are obtained by analytic, probabilistic and combinatoric tools. The first chapter deals with the strong polarization problem. This states that for any sequence $u_1,\dots, u_n$ of norm 1 vectors in a real Hilbert space $\mathscr H$, there exists a unit vector $v \in \mathscr H$, such that $$ \sum \frac{1}{\langle u_i, v \rangle^2} \leq n^2. $$ The 2-dimensional case is proved by complex analytic methods. For the higher dimensional extremal cases, we prove a tensorisation result that is similar to F. John's theorem about characterisation of ellipsoids of maximal volume. From this, we deduce that the only full dimensional locally extremal system is the orthonormal system. We also obtain the same result for the weaker, original polarization problem. The second chapter investigates a problem in probabilistic geometry. Take $n$ independent, uniform random points in a triangle $T$. Convex chains between two fixed vertices of $T$ are defined naturally. Let $L_n$ denote the maximal size of a convex chain. We prove that the expectation of $L_n$ is asymptotically $\alpha \, n^{1/3}$, where $\alpha$ is a constant between 1.5 and 3.5 -- we conjecture that the correct value is 3. We also prove strong concentration results for $L_n$, which, in turn, imply a limit shape result for the longest convex chains.