# A function that is convex but not strictly convex

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.

• Try all $c_i=0$. – max_zorn Apr 19 at 4:49
• Thanks!!! It worked :) – diego reyes Apr 23 at 3:19