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 $\M_\D$.