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:


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


 * 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 defined as
By setting $\alpha = t$ and $\beta = 1 - t$, this can also be written as:


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

Also see

 * Equivalence of Definitions of Convex Real Function