Let $f,g : \mathbb{R}^n \to [0,\infty)$ be two convex functions such that $f(x) = \left|p \cdot x \right|$ for some $p \in \mathbb{R}^n$ ( the absolute value of the dot product between $p$ and $x$) and $g(x) = \left\| x\right\|_2$ denote the $l_2$ norm of $x$.
I wish to show whether $ h(x) = \min \left\lbrace f(x), g(x) \right\rbrace$ is convex function. For such function ($h(x)$), if i compute the second derivative to check whether the Hessian is positive semi-definite then i get that its always zero yet i am not sure of this result.
Please advise.