Definition:Concave Real Function

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Definition

Let $f$ be a real function which is defined on a real interval $I$.


Definition 1

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

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


Definition 2

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

$\displaystyle \forall x_1, x_2, x_3 \in I: x_1 < x_2< x_3: \frac {f \left({x_2}\right) - f \left({x_1}\right)} {x_2 - x_1} \ge \frac {f \left({x_3}\right) - f \left({x_2}\right)} {x_3 - x_2}$


Definition 3

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

$\forall x_1, x_2, x_3 \in I: x_1 < x_2< x_3: \dfrac {f \left({x_2}\right) - f \left({x_1}\right)} {x_2 - x_1} \ge \dfrac {f \left({x_3}\right) - f \left({x_1}\right)} {x_3 - x_1}$


Also known as

A concave function can also be referred to as:

  • a concave down function
  • a convex up function


Also see

  • Results about concave real functions can be found here.