Definition:Concave Real Function/Definition 1

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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:

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

Geometric Interpretation

Let $f$ be a concave real function.


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