Exercise 22, § 4

Let $f_1, \ldots, f_n$ be continuous convex functions on a compact interval $I \subset \mathbf{R}$; suppose that for all $x \in I$ one has $\sup(f_j(x)) \geq 0$. Show that there exist $n$ numbers $\alpha_j \geq 0$ such that $\sum_{j=1}^n \alpha_j = 1$ and that $\sum_{j=1}^n \alpha_j f_j(x) \geq 0$ on $I$. (First treat the case $n = 2$, considering a point $x_0$ where the upper envelope $\sup(f_1, f_2)$ attains its minimum, when $x_0$ is interior to $I$ determine $\alpha_1$ so that the left derivative of $\alpha_1 f_1 + (1 - \alpha_1) f_2$ is zero at $x_0$. Pass to the general case by induction on $n$; use the induction hypothesis for the restrictions of $f_1, \ldots, f_{n-1}$ to the compact interval where $f_n(x) \leq 0$.)