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:

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

Strictly Concave

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

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

Geometric Interpretation

Let $f$ be a concave real function.

Then:

for every pair of points $A$ and $B$ on the graph of $f$, the line segment $AB$ lies entirely below the graph.

Also known as

A concave function can also be referred to as:

a concave down function

a convex up function.

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): Chapter $1$ Introduction: $1.7$: Terminology and Notation
• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 12$