Let $\mathcal{X}$ be an SNC model of $X$ and let $\mathfrak{a}$ be a vertical fractional ideal sheaf on $\mathcal{X}$. Then $\varphi:=\log|\mathfrak{a}|\in\mathcal{D}(X)$ satisfies:

• (i)

  $\varphi\circ\emb_{\mathcal{X}}$ is piecewise affine and convex on each face of $\Delta_{\mathcal{X}}$;

• (ii)

  $\varphi\leq\varphi\circ p_{\mathcal{X}}$.