We present new structures and results on the set $𝓜 _𝒟 $ of mean functions on a given symmetric domain 𝒟 in ℝ². First, we construct on $𝓜 _𝒟 $ a structure of abelian group in which the neutral element is the 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 the closed ball with center the arithmetic mean and radius 1/2. We show in particular that the geometric and harmonic means lie on the boundary of $𝓜 _𝒟 $. Finally, we give two theorems generalizing the construction of the AGM mean. Roughly speaking, those theorems show that for any two given means M₁ and M₂, which satisfy some regularity conditions, there exists a unique mean M satisfying the functional equation M(M₁,M₂) = M.