Let us say that a continuous metric on a line bundle $L$ is semi-positive if it is the uniform limit of a sequence of smooth semi-positive metrics on the same line bundle $L$. As in the complex case, we then say that a metrized line bundle is admissible if it can be written as $\overline{L}\otimes\overline{M}^{\vee}$, for two line bundles $L$ and $M$ with continuous semi-positive metrics.

Let $L$ be a metrized line bundle, and let $\left\|{\cdot}\right\|_{1}$ and $\left\|{\cdot}\right\|_{2}$ be two continuous metrics on $L$. It follows from the definition that the metrics $\left\|{\cdot}\right\|_{\min}=\min(\left\|{\cdot}\right\|_{1},\left\|{\cdot}\right\|_{2})$ and $\left\|{\cdot}\right\|_{\max}=\max(\left\|{\cdot}\right\|_{1},\left\|{\cdot}\right\|_{2})$ are continuous metrics.

Moreover, these metrics $\left\|{\cdot}\right\|_{\min}$ and $\left\|{\cdot}\right\|_{\max}$ are smooth if the initial metrics are smooth. Indeed, there exists a model $\mathfrak{X}$, as well as two line bundles $\mathfrak{L}_{1}$ and $\mathfrak{L}_{2}$ extending the same power $L^{e}$ of $L$ and defining the metrics $\left\|{\cdot}\right\|_{1}$ and $\left\|{\cdot}\right\|_{2}$ respectively. We may assume that $\mathfrak{L}_{1}$ and $\mathfrak{L}_{2}$ have regular global sections $s_{1}$ and $s_{2}$ on $\mathfrak{X}$ which coincide on $X$, with divisors $\mathfrak{D}_{1}$ and $\mathfrak{D}_{2}$ respectively. (The general case follows, by twisting $\mathfrak{L}_{1}$ and $\mathfrak{L}_{2}$ by a sufficiently ample line bundle on $\mathfrak{X}$.) The blow-up $\pi\colon\mathfrak{X}^{\prime}\rightarrow\mathfrak{X}$ of the ideal $\mathfrak{I}_{\mathfrak{D}_{1}}+\mathfrak{I}_{\mathfrak{D}_{2}}$ ; it carries an invertible ideal sheaf $\mathfrak{I}_{\mathfrak{E}}=\pi^{*}(\mathfrak{I}_{\mathfrak{D}_{1}}+\mathfrak{I}_{\mathfrak{D}_{2}})$, with corresponding Cartier divisor $\mathfrak{E}$. Since $\mathfrak{D}_{1}$ and $\mathfrak{D}_{2}$ coincide on the generic fiber, $\mathfrak{I}_{\mathfrak{D}_{1}}+\mathfrak{I}_{\mathfrak{D}_{2}}$ is already invertible there and $\pi$ is an isomorphism on the generic fiber.

The divisors $\pi^{*}\mathfrak{D}_{1}$ and $\pi^{*}\mathfrak{D}_{2}$ decompose canonically as sums

$$\pi^{*}\mathfrak{D}_{1}=\mathfrak{D}^{\prime}_{1}+\mathfrak{E},\quad\pi^{*}\mathfrak{D}_{2}=\mathfrak{D}^{\prime}_{2}+\mathfrak{E}.$$

Let us pose

$$\mathfrak{D}^{\prime}=\mathfrak{D}^{\prime}_{1}+\mathfrak{D}^{\prime}_{2}+\mathfrak{E}=\mathfrak{D}^{\prime}_{1}+\pi^{*}\mathfrak{D}_{2}=\pi^{*}\mathfrak{D}_{1}+\mathfrak{D}^{\prime}_{2}.$$

An explicit computation on the blow-up shows that $(\mathfrak{X}^{\prime},\mathfrak{D}^{\prime},e)$ and $(\mathfrak{X}^{\prime},\mathfrak{E},e)$ are models of $\left\|{\cdot}\right\|_{\min}$ and $\left\|{\cdot}\right\|_{\max}$ respectively. In particular, these metrics are smooth.

Assume that the initial metrics are semi-positive, and that some positive power of $L$ is effective. Then, the metric $\left\|{\cdot}\right\|_{\min}$ is semi-positive too. By approximation, it suffices to treat the case where the initial metrics are smooth and semi-positive. Then, the previous construction applies. Keeping the introduced notation, let us show that the restriction to the special fiber of the divisor $\mathfrak{(}D^{\prime})_{\tilde{K}}$ is numerically effective. Let $C\subset\mathfrak{X}^{\prime}_{\tilde{K}}$ be an integral curve and let us prove that $C\cdot\mathfrak{(}D^{\prime})_{\tilde{K}}$ is nonnegative. If $C$ is not contained in $\mathfrak{D}^{\prime}_{1}$, then $C\cdot(\mathfrak{D}^{\prime}_{1})_{\tilde{K}}\geq 0$, and $C\cdot(\pi^{*}\mathfrak{D}_{2})_{\tilde{K}}=\pi_{*}C\cdot\mathfrak{D}_{2}\geq 0$ since $(\mathfrak{D}_{2})_{\tilde{K}}$ is numerically effective ; consequently, $C\cdot\mathfrak{(}D^{\prime})_{\tilde{K}}\geq 0$. Similarly, $C\cdot\mathfrak{(}D^{\prime})_{\tilde{K}}\geq 0$ when $C$ is not contained in $\mathfrak{D}^{\prime}_{2}$. Since $\mathfrak{D}^{\prime}_{1}\cap\mathfrak{D}^{\prime}_{2}=\emptyset$, this shows that $C\cdot\mathfrak{D}^{\prime}_{\tilde{K}}\geq 0$ in any case, hence $(\mathfrak{D}^{\prime})_{\tilde{K}}$ is numerically effective.

This last result is the analogue in the ultrametric case to the fact that the maximum of two continuous plurisubharmonic functions is continuous plurisubharmonic. However, observe that in the complex case, the maximum or the minimum of smooth functions are not smooth in general.