Definition:Convex Real Function
(Redirected from Definition:Concave Up Function)
Jump to navigation
Jump to search
This page is about Convex Real Function. For other uses, see Convex.
Definition
Let $f$ be a real function which is defined on a real interval $I$.
Definition 1
$f$ is convex on $I$ if and only if:
- $\forall x, y \in I: \forall \alpha, \beta \in \R_{>0}, \alpha + \beta = 1: \map f {\alpha x + \beta y} \le \alpha \map f x + \beta \map f y$
Definition 2
$f$ is convex on $I$ if and only if:
- $\forall x_1, x_2, x_3 \in I: x_1 < x_2 < x_3: \dfrac {\map f {x_2} - \map f {x_1} } {x_2 - x_1} \le \dfrac {\map f {x_3} - \map f {x_2} } {x_3 - x_2}$
Definition 3
$f$ is convex on $I$ if and only if:
- $\forall x_1, x_2, x_3 \in I: x_1 < x_2 < x_3: \dfrac {\map f {x_2} - \map f {x_1} } {x_2 - x_1} \le \dfrac {\map f {x_3} - \map f {x_1} } {x_3 - x_1}$
Geometric Interpretation
Let $f$ be a convex real function.
Then:
- for every pair of points $A$ and $B$ on the graph of $f$, the line segment $AB$ lies entirely above the graph.
Also known as
A convex function can also be referred to as:
Examples
Square Function
The square function:
- $\forall x \in \R: \map f x = x^2$
is a convex real function.
Also see
- Results about convex real functions can be found here.