The centroidal mean of two numbers $a,b$ is the number $\dfrac{2(a^2+ab+b^2)}{3(a+b)}.$ In a trapezoid whose bases have lengths $a$ and $b$, it is the length of the line segment parallel to the bases which passes through the centroid.
For $p\in \Bbb R\setminus 0,$ a generalized $p$-mean of two numbers $a,b$ is the number $\left(\dfrac{a^p+b^p}2\right)^{1/p}.$
Is the centroidal mean a generalized $p$-mean for some $p$? I suspect it isn't, but I have no idea how to prove it. Just trying to equate the two expressions has led me nowhere.