Narrow Results

The purpose of this note is to study the asymptotic volume of intersections of unit balls associated with two norms in ${ℝ}^n$ 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.

In this paper, we prove a multivariate central limit theorem for ${\ell }_q$-norms of high-dimensional random vectors that are chosen uniformly at random in an ${\ell }_p^n$-ball. As a consequence, we provide several applications on the intersections of ${\ell }_p^n$-balls in the flavor of Schechtman and Schmuckenschläger and obtain a central limit theorem for the length of a projection of an ${\ell }_p^n$-ball onto a line spanned by a random direction $𝜃\in {𝕊}^{n-1}$. The latter generalizes results obtained for the cube by Paouris, Pivovarov and Zinn and by Kabluchko, Litvak and Zaporozhets. Moreover, we complement our central limit theorems by providing a complete description of the large deviation behavior, which covers fluctuations far beyond the Gaussian scale. In the regime $1\le p<q$ this displays in speed and rate function deviations of the $q$-norm on an ${\ell }_p^n$-ball obtained by Schechtman and Zinn, but we obtain explicit constants.