Given a list of $N$ real numbers $s_i$:
the median $x_0$ is
$$ \arg \min_{x_0} \sum_{i = 1}^{N} \left| {s}_{i} - x_0 \right| $$
the mean $x_2$ is
$$ \arg \min_{x_2} \sum_{i = 1}^{N} \left( {s}_{i} - x_2 \right)^2 $$
What about $x_4$?
$$ \arg \min_{x_4} \sum_{i = 1}^{N} \left( {s}_{i} - x_4 \right)^4 $$
does have it a name?
And what about the generic $x_{2n}$
$$ \arg \min_{x_{2n}} \sum_{i = 1}^{N} \left( {s}_{i} - x_{2n} \right)^{2n} $$
Are there any known application for $n>1$?
They are partial moments with respect to the minimizing reference point of an un-normalized uniformly distributed probability function. From wikipedia: $$\mu_n^-(r) + \mu_n^+(r)$$
$$r_o = \min_{r}\left\{\int_{-\infty}^\infty (r-x)^n f(x)dx\right\}$$
Where $f(x)$ is uniform discrete probability density $[0,N]$. (Since we are working with continuous case and integral instead of sum this is actually a sequence of Paul Dirac's delta distribution at integer values.)
Edit I first misread as $(x_k-i)^k$, that would have given the uniform discrete distribution mentioned above. Now what instead happens is we have sum of Dirac distributions like this: $$f(x) = \sum_{i=0}^N \delta(x-s_i)$$