Quasi-arithmetic means ad libitum

Abstract

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