Quasi-Arithmetic Means Ad Libitum

Abstract

AbstractLet $$\alpha _1, \ldots , \alpha _m$$ α 1 , … , α m be two or more positive reals with sum 1, let $$C\subseteq {\mathbb {R}}^k$$ C ⊆ R k be an open convex set, and $$f: C\rightarrow {\mathbb {R}}^k$$ f : C → R k be a continuous injection with convex image. For each nonempty set $$S\subseteq C$$ S ⊆ C , let $${\mathscr {M}}(S)$$ M ( S ) be the family of quasi-arithmetic means of all m-tuples of vectors in C with respect to f and the weights $$\alpha _1,\ldots ,\alpha _m$$ α 1 , … , α m , that is, the family $$\begin{aligned} {\mathscr {M}}(S)= \left\{ f^{-1}\left( \alpha _1f(x_1)+\cdots +\alpha _mf(x_m)\right) : x_1,\ldots ,x_m \in S \right\} . \end{aligned}$$ M ( S ) = f - 1 α 1 f ( x 1 ) + ⋯ + α m f ( x m ) : x 1 , … , x m ∈ S . We provide a simple necessary and sufficient condition on S for which the infinite iteration $$\bigcup _{n}{\mathscr {M}}^n(S)$$ ⋃ n M n ( S ) is relatively dense in the convex hull of S.