For all $p \geq 1$, $p$-norm is defined as, $$\Vert{x}\Vert_p=\left(\sum_{i=1}^{n} \vert{x_i}\vert^p\right)^{1/p}.$$
In this post, it has been shown that "every $p$-norm function is convex" for $x\in\Bbb{R}^n$. I failed to prove for $x\in\Bbb{Z}^n$, which results in "every $p$-norm function is sub-modular."
