Quasi-Arithmetic Means Ad Libitum

Abstract

Let alpha _1, ldots , alpha _m be two or more positive reals with sum 1, let Csubseteq {mathbb {R}}^k be an open convex set, and f: Crightarrow {mathbb {R}}^k be a continuous injection with convex image. For each nonempty set Ssubseteq 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 M(S)=f-1α1f(x1)+⋯+αmf(xm):x1,…,xm∈S.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\mathscr {M}}(S)= \\left\\{ f^{-1}\\left( \\alpha _1f(x_1)+\\cdots +\\alpha _mf(x_m)\\right) : x_1,\\ldots ,x_m \\in S \\right\\} . \\end{aligned}$$\\end{document}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.