Weak continuity on the variation of Newton Okounkov bodies

We start by presenting new tools and results suitable for
the study of valuations of higher rank on function fields of algebraic
varieties. This will be based on a study of higher rank quasi-monomial
valuations taking values in the lexicographically ordered group R^k.
This gives us a space of higher rank valuations that we endow with a
weak "tropical" topology. In this setting, we show that the Newton
Okounkov bodies of a given line bundle vary continuously with respect
to the valuation. We explain how this result fits in the literature
and how it gives us a restriction in the existence of mutations of
Newton Okounkov bodies. Joint work with Omid Amini.