Prove convexity of the given function on $\mathbb{R}^n$

Question

$f:\mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}$ is given by $$f(x)=\mbox{max}\{ (z_1-x_1)^{+}, (z_2-x_2)^{+}, ..., (z_3-x_3)^{+} \}$$ where $x=(x_2,x_2,...,x_n) \in \mathbb{R}^n$ and $z = (z_1,z_2,..,z_n)\in \mathbb{R}^n$ and $(z_i-x_i)^{+}=\mbox{max}\{ z_i-x_i,0 \}$.

Prove that $f(x)$ is convex for any fix $z\in \mathbb{R}^n$.

Answer 1

This is $f(x)=\max\{z_1-x_1,z_2-x_2,\cdots, z_n-x_n,0\}$. Supremum of a family, finite or otherwise, of convex functions is convex, and affine maps (meaning, maps in the form $g(x)=\langle b,x\rangle +\alpha$) are convex.