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](http://en.wikipedia.org/wiki/Lyapunov_condition) 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.



What m minimizes E(|m-X|^3) for a random variable X?
The minimizer $m$ is the nearest point projection of $X$ onto the subspace of $L^p$ formed by the constant functions ($p=3$ in your case). This $m$ is sometimes called the $p$-prediction or $p$-predictor of $X$. Apparently, this terminology began with [Andô and Amemiya](http://www.ams.org/mathscinet-getitem?mr=189077). Some of later papers are [Landers and Rogge](http://www.ams.org/mathscinet-getitem?mr=537095) (who wrote a few other papers, e.g. [this one](http://www.ams.org/mathscinet-getitem?mr=567995)), and [Cuesta and Matrán](http://www.ams.org/mathscinet-getitem?mr=894443). The term "generalized (conditional) expectation" also appeared.


