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

A couple weeks ago in a technical stretch of a proof involving the <a href="http://en.wikipedia.org/wiki/Lyapunov_condition">Lyapunov condition for the central limit theorem</a> I ended up with the expression E(|m-X|^3).  Does this statistic have a name, or any nice properties?

<b>Edit:</b> Earlier versions of this question had |m-EX| where |m-X| was; this isn't what I meant.