When $\dim(X)\geq 3$, there are varieties with line bundles $L$ such that $\vol(L)$ is irrational (see [ELMN05, Example 2.2] or [Laz04, Example 2.3.8]). Hence with our definition and normalization of non-archimedean volumes, we get for a model metric $\|\ \|$ that $\vol(L,\|\ \|,e^{\lambda}\|\ \|)=\vol(L)\lambda$ which produces irrational non-archimedean volumes. The following natural questions remain open:

1. (i)

   What happens in dimension two? Are non-archimedean volumes rational? Note that by Zariski decomposition, $\vol(L)$ is rational then (see for instance [Laz04, Cor. 2.3.22]).

2. (ii)

   If we normalize our non-archimedean volumes by $\vol(L)$, can we find an example of some model metrics $\|\ \|$ and $\|\ \|^{\prime}$ with irrational $\vol(L,\|\ \|,\|\ \|^{\prime})$? The idea is to avoid the trivial example above.