Definition:Convex Real Function/Definition 1

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

$f$ is convex on $I$ :


 * $\forall x, y \in I: \forall \alpha, \beta \in \R_{>0}, \alpha + \beta = 1: \map f {\alpha x + \beta y} \le \alpha \map f x + \beta \map f y$


 * ConvexFunction1.png

The geometric interpretation is that any point on the chord drawn on the graph of any convex function always lies on or above the graph.

Also see

 * Equivalence of Definitions of Convex Real Function