Limit Theorems for the Volume of Random Projections and Sections of $\ell_p^N$-balls

TL;DR

This paper establishes limit theorems for the volume and other geometric quantities of random projections and sections of high-dimensional $ ext{l}_p$-balls, providing a comprehensive asymptotic analysis as the dimension grows.

Contribution

It generalizes previous results by proving central limit theorems and deviation principles for various geometric measures of random projections and sections of $ ext{l}_p$-balls.

Findings

Proved central limit theorems for volume and intrinsic volumes.

Established moderate and large deviation principles.

Provided a complete asymptotic description of geometric quantities.

Abstract

Let $\mathbb{B}_p^N$ be the $N$-dimensional unit ball corresponding to the $\ell_p$-norm. For each $N\in\mathbb N$ we sample a uniform random subspace $E_N$ of fixed dimension $m\in\mathbb{N}$ and consider the volume of $\mathbb{B}_p^N$ projected onto $E_N$ or intersected with $E_N$. We also consider geometric quantities other than the volume such as the intrinsic volumes or the dual volumes. In this setting we prove central limit theorems, moderate deviation principles, and large deviation principles as $N\to\infty$. Our results provide a complete asymptotic picture. In particular, they generalize and complement a result of Paouris, Pivovarov, and Zinn [A central limit theorem for projections of the cube, Probab. Theory Related Fields. 159 (2014), 701-719] and another result of Adamczak, Pivovarov, and Simanjuntak [Limit theorems for the volumes of small codimensional random sections of $\ell_p^n$-balls, Ann. Probab. 52 (2024), 93-126].