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

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 ${\mathrm{\ell }}_p^n$-balls the focus lies on the case $p=\mathrm{\infty }$, 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 $p=\mathrm{\infty }$. In both situations we uncover an unconventional limiting behavior.