Quasi-Arithmetic Means Ad Libitum

Let alpha(1), ... , alpha(m) be two or more positive reals with sum 1, let C subset of R-k be an open convex set, and f : C -> R-k be a continuous injection with convex image. For each nonempty set S subset of 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 alpha(1), ... , alpha(m), that is, the family M(S) = {integral(-1) (alpha(1)integral(x(1)) + center dot center dot center dot + alpha(m)integral(x(m))) : x(1), ... , x(m) is an element of S}. We provide a simple necessary and sufficient condition on S for which the infinite iteration U-n M-n (S) is relatively dense in the convex hull of S.