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In this paper, we present new structures and results on the set MD of mean functions on a given symmetric domain D of R 2 . First, we construct on MD a structure of abelian group in which the neutral element is simply the Arithmetic mean; then we study some symmetries in that group. Next, we construct on MD a structure of metric space under which MD is nothing else the closed ball with center the Arithmetic mean and radius 1=2. We show in particular that the Geometric and Harmonic means lie in the border of MD . Finally, we give two important 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 regular conditions, there exists a unique mean M satisfying the functional equation: M(M1;M2) = M . MSC: 20K99; 54E35; 39B22; 39B12.

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