Let X be a random variable. Then E(|m-EX|^1) is minimized when (as a function of m) when m is the median of X, and E(|m-EX|^2) is minimized when m is the mean of x.

A couple weeks ago in a technical stretch of a proof involving the Lyapunov condition for the central limit theorem I ended up with the expression E(|m-EX|^3). Does this statistic have a name, or any nice properties?