Suppose that, $$ x^*=\underset{x\in\Bbb{Z}}{\operatorname{argmin}} \left \{ \Vert{x-a}\Vert_p \right\},$$
where $a \in \Bbb{Z}^n_+$ and,
$$ \Vert{x-a}\Vert_p=\left(\sum_{i=1}^{n} \vert{x - a_i}\vert^p\right)^{1/p}.$$
Can we say that $x^*$ is equal to the median of set $\left \{ a_i , \; \forall i \in 1..n\right \}$? If yes, is there any chance to prove it?
This is true for p = 1 (I leave the proof as an elementary exercise for you), but false in general for p > 1.
Consider the following example: a= (1 2 3 4 10).
Then the optimal x is respectively 3, 4, and 5 for p = 1, 2, and infinity.