Algebraic and topological structures on the set of mean functions and generalization of the AGM mean
Volume 132 / 2013
Abstract
We present new structures and results on the set ${\mathcal {M}}_\mathscr {D}$ of mean functions on a given symmetric domain $\mathscr {D}$ in $\mathbb {R}^2$. First, we construct on ${\mathcal {M}}_\mathscr {D}$ 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 ${\mathcal {M}}_\mathscr {D}$ a structure of metric space under which ${\mathcal {M}}_\mathscr {D}$ 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 ${\mathcal {M}}_\mathscr {D}$. Finally, we give two theorems generalizing the construction of the ${\rm AGM}$ mean. Roughly speaking, those theorems show that for any two given means $M_1$ and $M_2$, which satisfy some regularity conditions, there exists a unique mean $M$ satisfying the functional equation $M(M_1 , M_2) = M$.
