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 $I$ iff:


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

Equivalently:


 * $\forall x, y \in I: \forall t \in \left({0 \,.\,.\, 1}\right) : f \left({tx + \left({1 - t}\right) y}\right) \le t f\left({x}\right) + \left({1 - t}\right) f \left({y}\right)$

The function $f$ is strictly convex on $I$ if, in the above inequalities, equality cannot hold unless $x = y$.


 * 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