Exercise 19, § 1

Let $f$ be a convex function on an interval $I$, let $h$ be a number $> 0$, and $I_h$ the intersection of $I$ and the intervals $I + h$ and $I - h$; show that, if $I_h$ is not empty, the function

is convex on $I_h$; if $h < k$ one has $g_h \leq g_k$. When $h$ tends to 0 show that $g_h$ tends uniformly to $f$ on every compact interval contained in the interior of $I$.