Axiomatic approach to means

Question

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 mean as a function $f:({\mathbb R}_{>0})^n \to {\mathbb R}$ which satisfies the following:

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

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

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

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

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

Answer 1

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

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

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

Answer 2

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)

Answer 3

Question 1 asks for axiomatic approaches to means. Let me quote something:

The theory of means took shape in the first half of the twentieth century, with the 1930 papers of Kolmogorov [190, 192] and Nagumo [257] as well as Hardy, Littlewood and Pólya’s seminal book Inequalities [135], first published in 1934. (Aczél [1] describes the early history.) But new results continue to be proved. The 2009 book by Grabisch, Marichal, Mesiar and Pap lists some modern developments ([124], Chapter 4), and most of the characterization theorems in this chapter also appear to be new.

This is from the introduction to Chapter 5 of my own book Entropy and Diversity: The Axiomatic Approach (Cambridge University Press, 2021). Theorem 5.3.2 there states:

Theorem Let $(M: (0, \infty)^n \to (0, \infty))_{n \geq 1}$ be a sequence of functions. The following are equivalent:

• $M$ is homogeneous, strictly increasing, symmetric and decomposable;

• $M$ is the power mean $(x_1, \ldots, x_n) \mapsto \Bigl( \sum x_i^t \Bigr)^{1/t}$ for some $t \in \mathbb{R}$.

"Homogeneous", "strictly increasing" and "symmetric" are your conditions (ii)-(iv). "Decomposable" is the condition that $$ M(x_{1,1}, \ldots, x_{1,k_1}, \ \ldots,\ x_{n,1}, \ldots, x_{n, k_n}) = M(a_1, \ldots, a_1, \ \ldots,\ a_n, \ldots, a_n), $$ where $a_i = M(x_{i, 1}, \ldots, x_{i, k_i})$ and there are $k_i$ copies of $a_i$ on the right-hand side. Decomposability is very powerful, and in its absence, I suspect there are many other "means".

Answer 4

The following paper gives an axiomatization of the geometric mean, and suggests extensions to symmetric matrices:

Ando, T.; Li, Chi-Kwong; Mathias, Roy, Geometric means, Linear Algebra Appl. 385, 305-334 (2004). ZBL1063.47013.