SCHWEITZER 1963 Problem 7

Prove that for every convex function $f(x)$ defined on the interval $-1\leq x \leq 1$ and having absolute value at most $1$ , there is a linear function $h(x)$ such that $$\int_{-1}^1|f(x)-h(x)|dx\leq 4-\sqrt{8}.$$ [L. Fejes-Toth]