[Paper Review] Analytic and Probabilistic Problems in Discrete Geometry

Gergely Ambrus|arXiv (Cornell University)|Jul 11, 2019

Point processes and geometric inequalities40 references19 citations

TL;DR

This thesis resolves two central problems in discrete geometry using analytic, probabilistic, and combinatorial methods. It proves the strong polarization problem in 2D via complex analysis and extends it to higher dimensions using tensorization and John-type theorems, showing orthonormal systems are the only extremal configurations. In probabilistic geometry, it establishes that the expected length of the longest convex chain in a random triangle is asymptotically $\alpha n^{1/3}$ with $\alpha \approx 3$, and proves strong concentration and limit shape results.

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 $$ \sum \frac{1}{\langle u_i, v angle^2} \leq n^2. $$ 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 $α\, n^{1/3}$, where $α$ 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.

Motivation & Objective

• To resolve the strong polarization problem in real Hilbert spaces using analytic and geometric tools.
• To characterize extremal vector systems for the polarization problem, showing orthonormal systems are the only full-dimensional extremal configurations.
• To analyze the asymptotic behavior of the longest convex chain in a triangle with $n$ i.i.d. uniform random points.
• To establish strong concentration and limit shape results for the length of the longest convex chain.
• To numerically investigate the constant $\alpha$ in the asymptotic $\mathbb{E}[L_n] \sim \alpha n^{1/3}$, conjecturing $\alpha = 3$.

Proposed method

• Uses complex analytic methods, particularly equioscillating functions, to prove the strong polarization problem in 2D.
• Applies a tensorization argument analogous to F. John’s theorem to extend results to higher dimensions.
• Employs linear algebraic transformations to relate the problem to inverse eigenvectors of Gram matrices.
• Derives geometric interpretations of the difference between the original and strong polarization conjectures.
• Applies probabilistic concentration inequalities to show $L_n$ is tightly concentrated around its mean.
• Uses a $O(n^2)$ algorithm with spatial and length-based pruning to simulate longest convex chains in random point sets.

Experimental results

Research questions

• RQ1 What is the optimal bound for $\sum_{i=1}^n \frac{1}{\langle u_i, v \rangle^2}$ over unit vectors $u_i$ and unit vectors $v$ in a Hilbert space?
• RQ2 Which vector systems achieve extremality in the polarization problem, and are orthonormal systems the only full-dimensional extremal configurations?
• RQ3 What is the asymptotic growth rate of the expected length $\mathbb{E}[L_n]$ of the longest convex chain between two vertices of a triangle with $n$ i.i.d. uniform random points?
• RQ4 How concentrated is $L_n$ around its mean, and does a limit shape emerge for the longest convex chains?
• RQ5 What is the numerical value of the constant $\alpha$ in $\mathbb{E}[L_n] \sim \alpha n^{1/3}$, and is $\alpha = 3$?

Key findings

• The strong polarization problem holds in 2D, proven via complex analytic techniques involving equioscillating functions.
• In higher dimensions, the only full-dimensional locally extremal system for the strong polarization problem is the orthonormal system.
• The expected length of the longest convex chain satisfies $\mathbb{E}[L_n] \sim \alpha n^{1/3}$ with $\alpha \in [1.5, 3.5]$, and numerical evidence supports $\alpha = 3$.
• The random variable $L_n$ exhibits strong concentration: $\mathbb{P}(|L_n - \mathbb{E}[L_n]| > b) < n^{-\gamma^2/14}$ for $\gamma \geq 1$.
• Simulations confirm that $n^{-1/3}\mathbb{E}[L_n]$ increases with $n$, and for $n = 10^6$, it reaches $2.976$, supporting $\alpha \approx 3$.
• The distribution of $L_n$ is highly concentrated, with small standard deviation even at $n = 10^6$, and the longest chains lie in a narrow neighborhood of width $O(n^{-1/3})$.