Convexity is clear from the definition. To prove stability under maxima, let $\varphi_{1},\varphi_{2}\in\mathcal{D}(X)$ be $\theta$-psh, pick a common determination $\mathcal{X}$ of $\theta$ and the $\varphi_{i}$’s and let $D_{i}\in\Div_{0}(\mathcal{X})_{\mathbf{Q}}$ be a representative of $\varphi_{i}$ for $i=1,2$.

Since the ample cone of $N^{1}(\mathcal{X}/S)$ is open, we may find ample line bundles $\mathcal{A}_{1},\dots,\mathcal{A}_{r}\in\Pic(\mathcal{X})$ whose numerical classes $\alpha_{1},\dots,\alpha_{r}$ form a basis of $N^{1}(\mathcal{X}/S)$. We may thus find $t_{1},\dots,t_{r}\in\mathbf{R}$ such that $\mathcal{L}:=\sum_{j}t_{j}\mathcal{A}_{j}$ is a representative of $\theta$ in $\Pic(\mathcal{X})_{\mathbf{R}}$. Let $\varepsilon_{1},\dots,\varepsilon_{r}>0$ be (small) positive numbers such that $t_{j}+\varepsilon_{j}\in\mathbf{Q}$ for each $i$ and set $\mathcal{L}_{\varepsilon}:=\sum_{j}(t_{j}+\varepsilon_{j})\mathcal{A}_{j}$. Since $\varphi,\varphi^{\prime}$ are $\theta$-psh it follows that $\mathcal{L}_{\varepsilon}+D_{i}$ is an ample $\mathbf{Q}$-divisor on $\mathcal{X}$ for $i=1,2$. We may thus find a positive integer $m$ such that $m\mathcal{L}_{\varepsilon}\in\Pic(\mathcal{X})$, $mD_{i}\in\Div_{0}(\mathcal{X})$ and both sheaves $\mathcal{O}_{\mathcal{X}}\left(m\left(\mathcal{L}_{\varepsilon}+D_{i}\right)\right)$, $i=1,2$ are generated by their global sections on $\mathcal{X}$. If we introduce the vertical fractional ideal sheaf

$$\mathfrak{a}_{m}:=\mathcal{O}_{\mathcal{X}}(mD_{1})+\mathcal{O}_{\mathcal{X}}(mD_{2})$$

then it follows that $\mathcal{O}_{\mathcal{X}}(m\mathcal{L}_{\varepsilon})\otimes\mathfrak{a}_{m}$ is also generated by its global sections. By Lemma 5.6, $\log|\mathfrak{a}_{m}|=m\max\left\{\varphi_{1},\varphi_{2}\right\}$ is thus psh with respect to $m(\theta+\sum_{j}\varepsilon_{j}\alpha_{j})$, that is:

$$\theta+\sum_{j}\varepsilon_{j}\alpha_{j}+dd^{c}\max\{\varphi_{1},\varphi_{2}\}\geq 0.$$

Letting $\varepsilon_{j}\to 0$, we conclude as desired that $\theta+dd^{c}\max\{\varphi_{1},\varphi_{2}\}\geq 0$. ∎