Generalization of vector-valued QAM characterization to full-dimensional closed convex domains

Prove an analogue of Theorem 4.2 for every closed convex set C contained in a finite-dimensional Euclidean space whose interior is nonempty, characterizing the strictly increasing generators whose vector-valued weighted quasi-arithmetic mean coincides with the corresponding weighted arithmetic mean.

Background

Theorem 4.2 characterizes the generators of vector-valued weighted quasi-arithmetic means that equal the weighted arithmetic mean when the domain is the entire Euclidean space. The concluding section shows that this conclusion fails for arbitrary closed convex domains by constructing a counterexample on a line segment, a domain with empty interior.

The authors explicitly ask whether the theorem can be extended to closed convex domains with nonempty interior, which excludes the pathological lower-dimensional example while retaining substantially more general domains than the whole space.