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 Lyapunov condition for the central limit theorem I ended up with the expression E(|m-X|^3). Does this statistic have a name, or any nice properties?
Edit: Earlier versions of this question had |m-EX| where |m-X| was; this isn't what I meant.