Algebraic and topological structures on the set of mean functions and generalization of the AGM mean

TL;DR

This paper explores algebraic and topological structures on the set of mean functions, introduces new group and metric space frameworks, and generalizes the AGM mean through two key theorems.

Contribution

It constructs a novel abelian group and metric space structure on mean functions and extends the AGM mean with two generalization theorems.

Findings

Mean functions form an abelian group with the arithmetic mean as neutral element.

The set of mean functions forms a closed ball in a metric space with radius 1/2.

Geometric and Harmonic means lie on the boundary of the set of mean functions.

Abstract

In this paper, we present new structures and results on the set $\M_\D$ of mean functions on a given symmetric domain $\D$ of $\mathbb{R}^2$. First, we construct on $\M_\D$ a structure of abelian group in which the neutral element is simply the {\it Arithmetic} mean; then we study some symmetries in that group. Next, we construct on $\M_\D$ a structure of metric space under which $\M_\D$ is nothing else the closed ball with center the {\it Arithmetic} mean and radius 1/2. We show in particular that the {\it Geometric} and {\it Harmonic} means lie in the border of $\M_\D$. Finally, we give two important theorems generalizing the construction of the $\AGM$ mean. Roughly speaking, those theorems show that for any two given means $M_1$ and $M_2$, which satisfy some regular conditions, there exists a unique mean $M$ satisfying the functional equation: $M(M_1, M_2) = M$.