I've recently learned about the Generalized Mean as an abstraction of the many different means, includeing the Geometric, Arithmetic, and Harmonic means, as well as others.
It is my understanding that different means are more meaningful for different data sets, as far as providing the true Measure of Central Tendency.
My question is this: Is there a way to use the Generalized mean as a continuous (or discrete) function in it's variable $p$ - working with the same data set for all $p$ values of course, and from the output of that function, determine which value of $p$ would provide the best Measure of Central Tendency?
(I should add that if anyone has anything to add about the even more abstracted Generalized f-mean as it relates to this question that would be much welcomed as well)
First, if you use $f(x)=x^p$, then the $f$-mean is the same as the $p$-mean.
Second, "the best measure of central tendency" depends on the problem one is trying to solve much more than on the data set (although these aspects are of course related).
Third, a very important measure of central tendency - median - is not included in either generalization.