Recently I have been contemplating on a talk for high school children. One of my favorite topics in high school was the inequality of means. I had a great high school teacher who wrote some very nice articles (in Hebrew) about inequalities, so I was looking at some of them. This made me think about something I had wondered about when I was young, what does it mean a mean? Of course googling mean is not very useful. So I have two questions:

Do you know about any axiomatic approach to means?

Is it useful in anyway?

For instance, one could try and define a $\textbf{mean}$ as a function $f:({\mathbb R}_{>0})^n \to {\mathbb R}$ which satisfies the following:

(i) $\min_i\{x_i\} \leq f(x_1,x_2,\ldots,x_n) \leq \max_{i}\{x_i\}$.

(ii) $f(ax_1,ax_2,\ldots,ax_n)=af(x_1,x_2,\ldots,x_n)$.

(iii) $f$ is strictly monotone in each varaible.

(iv) If, in addition, $f$ is preserved by any permutation of the $x_i$'s, then we call it $\textbf{symmetric}$.

If $f$ is not symmetric, then one can define $G_f$, the $\textbf{group of symmetries of f}$, to be the symmetries that preserve $f$.