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

In this paper, we present new structures and results on the set ${\text{M}}_{\text{D}}$ of mean functions on a given symmetric domain $\text{D}$ of ${\mathbb{R}}^2$. First, we construct on ${\text{M}}_{\text{D}}$ 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 ${\text{M}}_{\text{D}}$ a structure of metric space under which ${\text{M}}_{\text{D}}$ 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 ${\text{M}}_{\text{D}}$. Finally, we give two important theorems generalizing the construction of the $\text{AGM}$ 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: $M(M_1,M_2)=M$.