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

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arxiv 2308.10635 v2 pith:NKPPRWAA submitted 2023-08-21 math.PR math.FA

classification math.PRmath.FA
keywords ballscasesclassicalinftyintersectionsnoteschattenassociated
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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.

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

    math.PR 2024-12 conditional novelty 7.0 of 10

    For fixed m, the rescaled volume of a random m-dimensional projection or section of an ell_p ball satisfies a CLT, an MDP, and an LDP with explicit limits as N grows.

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