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

Mathias Sonnleitner, Christoph Thäle

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.

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

Abstract

The purpose of this note is to study the asymptotic volume of intersections of unit balls associated with two norms in as their dimension tends to infinity. A general framework is provided and then specialized to the following cases. For classical -balls the focus lies on the case , which has previously not been studied in the literature. As far as Schatten -balls are considered, we concentrate on the cases and . In both situations we uncover an unconventional limiting behavior.
Paper Structure (4 sections, 12 theorems, 73 equations)

This paper contains 4 sections, 12 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. A general framework
  3. Proof of Theorem \ref{['thm:pball']}
  4. Proof of Theorem \ref{['thm:schatten-crit']}

Key Result

Proposition 1

Let $1\leq p\le \infty$ and $1\le q<\infty$ be such that $p\neq q$. Then there exists a constant $t_{p,q}\in (0,\infty)$ only depending on $p$ and $q$ such that

Theorems & Definitions (27)

  • Proposition 1: Schechtman/Schmuckenschläger
  • Remark 1
  • Theorem 2
  • Theorem 3
  • Remark 2
  • Remark 3
  • Conjecture 4
  • Lemma 5
  • proof
  • Lemma 6
  • ...and 17 more