Definition:Convex Real Function/Definition 3

Definition
Let $f$ be a real function which is defined on a real interval $I$.

$f$ is convex on $I$ iff:


 * $\forall x_1, x_2, x_3 \in I: x_1 < x_2 < x_3: \dfrac {f \left({x_2}\right) - f \left({x_1}\right)} {x_2 - x_1} \le \dfrac {f \left({x_3}\right) - f \left({x_1}\right)} {x_3 - x_1}$

Hence a geometrical interpretation: the slope of $P_1 P_2$ is less than or equal to that of $P_1 P_3$:


 * ConvexFunction3.png

Also see

 * Equivalence of Definitions of Convex Real Function