# the structure on the value group sort of a C-minimal field

Let $K$ be an algebraically closed valued field which is $C$-minimal, as defined, for example, in this article. Examples include pure algebraically closed valued fields, as well as Lipschitz and Robinson's expansions by rigid subanalytic sets (Lipshitz. Rigid subanalytic sets. Amer. J. Math., 115(1):77-108, 1993)

Is it true then that the structure induced on the value group of $K$ is that of a vector space (over a field that depends on $K$)? This would mean, in particular, that definable sets are Boolean combinations of sets defined by systems of affine inequalities (i.e. polyhedra).

I think the answer is no. Consider an algebraically closed valued field in the three sorted language. Using a relative quantifier elimination argument, any o-minimal expansion of the value group preserves $C$-minimality of the structure (meaning, definable subsets in one variable of the field sort are finite boolean combinations of balls). For instance, if your value group is the reals, putting the exponential function in the value group sort provides a counterexample.
Let me mention that a result of Cluckers in this article shows that your question is almost true for $P$-minimal expansions of $p$-adically closed fields where the 'almost' means that you need to allow congruence relations: the induced structure on the value group is precisely the Presburger structure.