Convexity of Norm of Max

Question

Let $p \ge 1$. Show the convexity of the function $h:\mathbb{R}^k \rightarrow \mathbb{R}$ defined as:

$$h(\textbf{z})=\left(\sum\limits_{i=1}^k \max\{z_i,0\}^p \right)^{1/p}$$

Answer 1

Consider whether, for a given $i$, $z_{i}$ and $w_{i}$ have the same or different sign. If they're the same, then it is trivial that $$\max\left\{\lambda z_{i}+(1-\lambda)w_{i},0\right\}=\lambda\max\left\{z_{i},0\right\}+(1-\lambda)\max\left\{w_{i},0\right\}$$ for all $\lambda\in[0,1]$. If they're different, then consider the inequality between $\lambda z_{i}$ and $(1-\lambda)w_{i}$.

To complete the proof, you might consider using the triangle property of a norm.