Exercise 22, § 1

Let $f$ be a numerical increasing continuous function and ≥ 0 in the interval (0, 1). (a) Now prove that there exists a convex function $g$ ≥ 0 in (0, 1) such that $g \leq f$ and $\int_0^1 g(t)\,dt \geq \frac{1}{2} \int_0^1 f(t)\,dt$ (cf. I, p. 53, exercise.