Berkovich spaces, polytopes and model theory

Let $X$ be a Berkovich analytic space and let $f= (f_{1},\ \ldots,\ f_{n})$ be a family of invertible holomorphic functions on it. We will explain how model theory can help to recover two results which were originally proven using de Jong's alterations, and which tell the following.

1. The image of $\log|f|=(\log|f_{1}|,\ \ldots,\ \log|f_{n}|)$ is a polytope if $X$ is compact.
2. If $n =\dim X$ the pre-image of the `skeletton' of $G_{m}^{n}$ under $f$ has a canonical piecewise-linear structure.

We will also explain how one can use model theory (and Temkin's reduction theory) to get a local version of a., telling that if $x$ is a point of $X$ and if $U$ is a small enough compact neighborhood of $x$, the germ of the polytope $\log|f|(U)$ at $\log|f|(x)$ doesn't depend on $U$.