Midpoint convex function

Remarks

Example: $f(x) = x^2$ on $\R$ is midpoint convex. Check: $\frac{x^2 + y^2}{2} - \left(\frac{x + y}{2}\right)^2 = \frac{(x - y)^2}{4} \ge 0$.

Non-example: $f(x) = -x^2$ on $\R$ is not midpoint convex, since $f(0) = 0 > -1 = \frac{f(-1) + f(1)}{2}$.

Every convex function is midpoint convex: take $t = 1/2$ in the definition of convexity. The converse is false in general. A discontinuous additive function $a \colon \R \to \R$ satisfies $a\big(\frac{x+y}{2}\big) = \frac{a(x) + a(y)}{2}$, so it is midpoint convex. But it is not convex, since convex functions on open intervals are continuous. A midpoint convex function that is continuous is convex.