A note on critical intersections of classical and Schatten $p$-balls

TL;DR

This paper investigates the asymptotic volume of intersections of classical and Schatten p-balls in high dimensions, revealing novel limiting behaviors especially for p=∞ and p=2 cases.

Contribution

It introduces a general framework for analyzing high-dimensional intersections of norm balls and explores previously unstudied cases for classical and Schatten p-balls.

Findings

Unconventional limiting behavior for p=∞ in classical ℓ_p^n-balls.

New asymptotic results for Schatten p-balls at p=2 and p=∞.

Extension of intersection volume analysis to high-dimensional normed spaces.

Abstract

The purpose of this note is to study the asymptotic volume of intersections of unit balls associated with two norms in $\mathbb{R}^n$ as their dimension $n$ tends to infinity. A general framework is provided and then specialized to the following cases. For classical $\ell_p^n$-balls the focus lies on the case $p=\infty$, which has previously not been studied in the literature. As far as Schatten $p$-balls are considered, we concentrate on the cases $p=2$ and $p=\infty$. In both situations we uncover an unconventional limiting behavior.