  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.
