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Let $V$ be a vector space over a field $k$. I was wondering if there is a standard terminology for a function $v: V \setminus \{0\} \to \mathbb{R}$ which is invariant under multiplication by nonzero scalars and satisfies non-Archimedean inequality $v(x+y) \geq \min\{v(x), v(y)\}$?

In the paper "Newton-Okounkov bodies, semigroups of intergral points, graded algebras and intersection theory", the authors call such a function a "pre-valuation", I was wondering if this concept has been introduced before and if there are other terms used for it.

Of course, this is related to the notion of ultra-metric.

Thanks

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