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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:

  1. Do you know about any axiomatic approach to means?

  2. 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 $f$.

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On the projective line, an important invariant is the cross-ratio (actually the only projective invariant of four points). Each of the three usual means, arithmetic, harmonic and geometric, are all instances of the cross-ratio. As an consequence, you can go from one mean to the other using an homography. I find it unexpected and I think this can be a way to introduce a bit of projective geometry. This also gives a geometric characterisation of the arithmetic mean amongst the other means.

Recall that four points A, B, C, D form an harmonic range if their cross-ratio is equal to -1. We choose a point at infinity on the projective line, together with an origin O and a unit point. Denote by a, b, c, d the coordinates of A, B, C, D on the line.

Denote the cross-ratio by $(a,b,c,d) = {(c-a)(d-b)\over (c-b)(d-a)}$.

If A is in O, then $(0,b,c,d)=-1$ and $2/b = 1/c + 1/d$ (harmonic mean).

If O is in the middle of AB, then $(a,-a,c,d)=-1$ and $a^2 = cd$ (geometric mean).

If A is at infinity , then $(\infty, b,c,d)=-1$ and $2b = c+d$ (arithmetic mean).

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The paper Social choice and topology a case of pure and applied mathematics by Beno Eckmann investigates what is meant by a mean on a topological space. In particular, if $k$ is a natural number, then they define a $k$-mean to be a continuous function $f:X^{k}\rightarrow X$ such that $f(x,...,x)=x$ and $f(x_{1},...,x_{k})=f(x_{\sigma(1)},...,x_{\sigma(k)})$ where $\sigma:\{1,...,k\}\rightarrow\{1,...,k\}$ is any permutation. (I should mention that I found out about this paper from this question)

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