Following a suggestion of Tony Yue Yu, we can define semipositivity locally on $V$. We say that a piecewise linear metric on $L$ is semipositive in $x\in V$ if there is a compact strictly $K$-analytic domain $W$ in $V$ which is a neighborhood of $x$ such that the restriction of ${\|\hskip 4.30554pt\|}$ to $L|_{W}$ is a semipositive formal metric in the sense of 3.2 (using the equivalence of Proposition 2.8). We say that ${\|\hskip 4.30554pt\|}$ is semipositive if it is semipositive in all $x\in V$. We will see in Proposition 3.10 that this fits with the definition in 3.2 assuming that $V$ is boundaryless.