Let $c_1,c_2,...,c_m\in\mathbb{R}^n$ and $b_1,b_2,...,b_m\in\mathbb{R}$. Consider $\mathbb{R}^n\ni x \mapsto f(x)=\displaystyle\max_{1\leq i\leq m}\Arrowvert c_i^Tx+b_i \Arrowvert$. Prove that $f$ is convex. Is $f$ strictly convex ? Strongly (uniformly) convex?
I have already proved that $f$ is indeed a convex function, my instinct says that $f$ is'nt a strictly and strongly convex function, but I don't find any counterexample to prove it.