A “mean”-space (in the sense of averaging) is defined by an invertible function $f(x)$, with the mean of a set of numbers $x_{1\dots n}$ being computed as $$ \langle x_\bullet \rangle_f = f^{-1}\left(\frac1n \sum_{k=1}^n f(x_k)\right). $$ In physics we generally require one further axiom on $f$: it must have a property of scale invariance so that for all $n$ and all scale-factors $s>0$, we expect the above to also equal $$\langle x_\bullet \rangle_f = s~f^{-1}\left(\frac1n \sum_{k=1}^n f(x_k / s)\right).$$
So for example $\log$ is a mean-space function, and you can thus compute an average in logarithmic space (it is the geometric mean), because on trying to scale all of the observations you get $$ s~\exp\left(-\log s + \frac1n \sum_{k=1}^n \log x_k\right) = \exp\left(\frac1n \sum_{k=1}^n \log x_k\right) = \langle x_\bullet \rangle_{\log}.$$ Other examples: in physics it is also popular to compute averages in reciprocal space $f(x) = 1/x$, and of course in linear space $f(x) = x$.
One could perhaps in certain circumstances compute averages in quadratic space $f(x) = x^2$; for example if one is computing the sum of independent Gaussian random variables then because variance is linear the resulting Gaussian has standard deviation $s = \sqrt{s_1^2 + s_2^2 + \dots + s_n^2},$ and one would have a mean of this form if considering $s/\sqrt{n}$ which would require a somewhat strange combination of the random variables, the variance-preserving combination $(X_1 + \dots + X_n)/\sqrt{n}$ as opposed to the mean-preserving combination $(X_1 + \dots + X_n)/n$. But maybe in certain circumstances where the means are 0 this is a helpful way to average “noises” together.
Counterexample: one cannot meaningfully average in $\exp$-space because it is scale-variant; the average of 1, 2, 3 (roughly 2.309) in exp-space is not half the average of 2, 4, 6 (roughly 5.044), which matters for physicists because our measurements do not result in single numbers but in equivalence classes of numbers related by scaling; you might have “1” meter but that is also “1000” millimeters and “0.001” kilometers and all of our mathematical operations must be well-defined on these equivalence classes.
My question is, does this notion of scale-invariance have any other interesting names or characterizations in mathematics? Like, does this class of functions have some sort of name or category-structure or so that has been studied before?