Maximum of convex functions

Question

how can i proof that: If $f_1, . . . , f_m$ are convex functions,than function $F(x) = \max(f_1(x), \dots , f_m(x))$ is convex? thanx for help.

Answer 1

Hint

Just prove that $$tF(x) + (1-t)F(y)\ge f_i(tx + (1-t)y)$$ for every $i \in \{1, \dots, m\}$ and $t \in [0, 1]$ by using the convexity of every $f_i$ and the definition of $F$ and then conclude.

Answer 2

Here is another way:

Let $\operatorname{epi} f = \{ (\alpha, x) | \alpha \ge f(x) \}$. Then $f$ is convex iff $\operatorname{epi} f$ is convex.

Note that $\operatorname{epi} F = \cap_k \operatorname{epi} f_k $, and since the intersection of convex sets is convex, it follows that $F$ is convex.

(The same technique shows that $f = \sup_\alpha f_\alpha$ is convex as well.)

This is from Rockafellar, an excellent text.