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][1]. Some of later papers are [Landers and Rogge][2] (who wrote a few other papers, e.g. [this one][2b]), and [Cuesta and Matrán][3]. The term "generalized (conditional) expectation" also appeared.

  [1]: http://www.ams.org/mathscinet-getitem?mr=189077
  [2]: http://www.ams.org/mathscinet-getitem?mr=537095
  [2b]: http://www.ams.org/mathscinet-getitem?mr=567995
  [3]: http://www.ams.org/mathscinet-getitem?mr=894443