Quasi-arithmetic means ad libitum

Let α1,...,αm be two or more positive reals with sum 1, let C ⊆ Rk be an open convex set, and f : C → Rk be a continuous injection with convex image. For each nonempty set S ⊆ C, let M(S) be the family of quasi-arithmetic means of all m-tuples of vectors in C with respect to f and the weights α1,...,αm, that is, the family M(S) = f −1 (α1f(x1) + ··· + αmf(xm)) : x1,...,xm ∈ S.
We provide a simple necessary and sufficient condition on S for which the infinite iterationn n Mn(S) is relatively dense in the convex hull of S.