Question 68
Show that, if $f: [0, \infty) \to \mathbb{R}$ is a convex function and $a, b, c$ are positive real numbers, then
$$
\begin{aligned}
& \int_{0}^{a} f(x) \, dx + \int_{0}^{b} f(x) \, dx + \int_{0}^{c} f(x) \, dx + \int_{0}^{a+b+c} f(x) \, dx \\
\ge \\
& \int_{0}^{a+b} f(x) \, dx + \int_{0}^{a+c} f(x) \, dx + \int_{0}^{b+c} f(x) \, dx.
\end{aligned}
$$
Dan Marinescu and Leonard Giugiuc
$$
\begin{aligned}
& \int_{0}^{a} f(x) \, dx + \int_{0}^{b} f(x) \, dx + \int_{0}^{c} f(x) \, dx + \int_{0}^{a+b+c} f(x) \, dx \\
\ge \\
& \int_{0}^{a+b} f(x) \, dx + \int_{0}^{a+c} f(x) \, dx + \int_{0}^{b+c} f(x) \, dx.
\end{aligned}
$$
Dan Marinescu and Leonard Giugiuc