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 $I$ if and only if:

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

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.

Strictly Concave

$f$ is strictly concave on $I$ if and only if:

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

Also see

• Equivalence of Definitions of Concave Real Function

Sources

• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 12.13$
• 1994: Martin J. Osborne and Ariel Rubinstein: A Course in Game Theory ... (previous) ... (next): $1.7$: Terminology and Notation
• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 12$