Quasi-arithmetic means ad libitum

TL;DR

This paper characterizes when iterated quasi-arithmetic means, defined via a continuous injection on a convex set, become dense in the convex hull of a set, providing a clear necessary and sufficient condition.

Contribution

It offers a simple criterion for the density of iterated quasi-arithmetic means in the convex hull, extending understanding of mean iteration behavior.

Findings

Identifies conditions for density of mean iterations

Provides a necessary and sufficient criterion

Extends theory of quasi-arithmetic means

Abstract

Let $\alpha_1, \ldots, \alpha_m$ 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 $\mathscr{M}(S)$ be the family of quasi-arithmetic means of all $m$-tuples of vectors in $C$ with respect to $f$ and the weights $\alpha_1,\ldots,\alpha_m$, that is, the family $$ \mathscr{M}(S)= \left\{ f^{-1}\left(\alpha_1f(x_1)+\cdots+\alpha_mf(x_m)\right): x_1,\ldots,x_m \in S \right\}. $$ We provide a simple necessary and sufficient condition on $S$ for which the infinite iteration $\bigcup_{n}\mathscr{M}^n(S)$ is relatively dense in the convex hull of $S$.