Let $\Omega$ be a measurable set having finite Lebesgue measure. Let $p\geq 1$ and $u\in L^p(\Omega)$. Is it true that the minimum value of the real function $$ c\in \mathbb{R}^n\mapsto\int_\Omega |u-c|^p \mathrm{d}\mu $$ is achieved when $c$ is the average of $u$? I'm interested in the case $p\neq 2$. Indeed for $p=2$ this is obvious.

By the way, if the average is no more a minimizer when $p\neq 2$, is it possible to find an explicit expression for a minimizer when $p\neq 2$?