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I was thinking if it's possible to generalize the idea of averages, like we have already the arithmetic, geometric and harmonic averages and I realized that they share some properties and I'd like to know if there's a theory about averages and generalizations.

So, my idea to try to tackle the problem is the following:

For each $n \in \mathbb{Z}_{>0} $ define the set $X_n = \{ (x_1, ..., x_n) \in \mathbb{R}^n : x_i > 0 \ \forall i \}$ . Consider the set $X = \cup_{n \in \mathbb{Z}_{>0}} X_n$.

Then what I realized is that averages satisfy the following properties:

Is a function $f: X \rightarrow \mathbb{R}_{>0}$

If we have $(x_1, ..., x_n) \in X$, let $m = \min \{x_1, ..., x_n\}$ and $M = \max \{x_1,...,x_n\}$ then $m \leq f(x_1, ..., x_n) \leq M$ with equality iff $x_1 = ... = x_n$ i.e. the "average" will always be strictly bigger than the minimum value and strictly less than the maximum if there are at least two different values

If $(x_1,...,x_n) \in X$ and $\sigma \in S_n$ the group of permutations of $\{ 1,..., n\}$ then $f(x_1,...,x_n) = f(x_{\sigma(1)}, ..., x_{\sigma(n)})$

If $(x_1,...,x_n) \in X$ and $x_{n+1} > 0$ then we have that:

$x_{n+1} > f(x_1,...,x_n) \Leftrightarrow f(x_1,...,x_n,x_{n+1}) > f(x_1,...,x_n)$ ,and we could also replace $>$ for $=$ or $<$

This last one is about pondered averages:

If we $(x_1,...,x_n) \in X$ then for every $m \in \mathbb{Z}_{>0}$ we have that $f(x_1,...,x_n) = f(y_1,...,y_m,...,y_{nm})$ where for each $k \in \{1,...,n \}$ we set $y_{(k-1)m + 1} =... = y_{km} = x_k$

So my question is if is there a way to generalize averages or if there's a theory corcening about it and if it can be interesting somehow

eipi10
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  • Maybe we should additionally require that $f$ is monotonic (in the first argument, say) and simplifiy the "$n+1$" condition to: $f(\vec x, f(\vec x))=f(\vec x)$. Also, why do you require the inputs to be positive? – Karl Feb 24 '21 at 00:58
  • The reason about positivity is that in general the famous averages use only positive values like the geometric average and the harmonic average. – eipi10 Feb 24 '21 at 01:00
  • Maybe also should put something like if we restrict the function to $X_n$ then the function is monotonic in every coordinate – eipi10 Feb 24 '21 at 01:05
  • Yeah, or just in the first coordinate (since we have the symmetry requirement). – Karl Feb 24 '21 at 01:07
  • Each of the averages you mentioned can be written as $f(x_1,\dots,x_n)=g^{-1}\left(\frac1n\left(g(x_1)+\dots+g(x_n)\right)\right)$ for some function $g$. It might be interesting to try establishing conditions (like the ones you provided) under which this must be the case. Maybe one can recover $g$ from $f$ somehow. – Karl Feb 24 '21 at 01:07
  • Just for curiousity which is the function $g$ for the geometric average? – eipi10 Feb 24 '21 at 01:15
  • Let us continue this discussion in chat. – eipi10 Feb 24 '21 at 01:53

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