Midpoint convex function

Summary

A midpoint convex function, also called a Jensen-convex function, is a real function $f$ on an interval with $f\big(\tfrac{x + y}{2}\big) \le \tfrac{f(x) + f(y)}{2}$ for all $x, y$. Every convex function is midpoint convex, but the converse fails without some regularity. Mild extra conditions close the gap: a midpoint convex function that is bounded above on a small interval, or that is measurable, is convex. The notion is the starting point of the theory of convex functions as solutions of a functional inequality.