Generalization of mean and median It is well known that a median of a distribution $\mu$ can be defined as an $m$ such that
$$m\in\operatorname*{arg\,min}_{c\in\mathbb{R}}\mathbb{E}_{X\sim\mu}[|X-c|].$$
Similarly, the mean of a distribution $\mu$ is defined as an $m$ such that
$$m=\operatorname*{arg\,min}_{c\in\mathbb{R}}\mathbb{E}_{X\sim\mu}[(X-c)^2].$$
I am interested in whether there is any reference or literature on the generalization of this to higher powers $p$ - in particular, what can be said about $m$ such that
$$m\in\operatorname*{arg\,min}_{c\in\mathbb{R}}\mathbb{E}_{X\sim\mu}[|X-c|^p]$$
for $p>2?$
Edit:
Coming back to this I believe the last line can be written as
$$m=\operatorname*{arg\,min}_{c\in\mathbb{R}}\mathbb{E}_{X\sim\mu}[|X-c|^p],$$
that is there is a unique minimizer (this is because for $p\in (1,\infty)$, $|\cdot|^p$ is strictly convex).
 A: 
I am interested in whether there is any reference or literature on the generalization of this to higher powers $p$

There's a "short communication" that defines the quantity you call $m$ as the "location parameter $L_p(\mathcal{D})$". It doesn't have all that many references, which seems to suggest that the authors came up with the idea on their own, rather than finding it in prior work:
Callegaro, L., & Pennecchi, F. (2007). Why always seek the expected value? A discussion relating to the Lp norm. Metrologia, 44(6), L68.

what can be said about $m$ for $p>2$?

The same two authors have an earlier paper in which they discuss estimators for $m$. This topic seems more widely studied and the references within this paper may worth looking at:
Pennecchi, F., & Callegaro, L. (2006). Between the mean and the median: the Lp estimator. Metrologia, 43(3), 213.
Most of the focus is on $1<p<2$, but there are some results for $p=\infty$ and generic $p$. For instance, the authors show how estimators of $m$ vary as a function of $p$ and the probability distribution. In particular, higher values of $p$ will give smaller estimator values for the uniform distribution. They also mention some asymptotic properties of estimators for $m$ (see Eqn 13 and surrounding text).
