Definition:Concave Real Function/Definition 1

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


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

wherever $x, y \in I$.


 * ConcaveFunction1.png

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

Also see

 * Equivalence of Definitions of Concave Real Function