Definition:Convex Real Function

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:

$\forall x_1, x_2, x_3 \in I: x_1 < x_2 < x_3: \dfrac {\map f {x_2} - \map f {x_1} } {x_2 - x_1} \le \dfrac {\map f {x_3} - \map f {x_1} } {x_3 - x_1}$

Convex at Point

Let $a \in I$ be a point in the domain of $f$.

Then:

$f$ is convex at $a$

if and only if:

there exists a neighborhood of $a$ on which $f$ is convex.

Vector Space

Let $\GF \in \set {\R, \C}$.

Let $X$ be a vector space over $\GF$.

Let $C \subseteq X$ be a convex set.

Let $f: C \to \overline \R$ be an extended real-valued function.

We say that $f$ is convex if and only if:

$\map f {t x + \paren {1 - t} y} \le t \map f x + \paren {1 - t} \map f y$ for each $t \in \openint 0 1$ and $x, y \in C$

whenever $\set {\map f x, \map f y} \ne \set {-\infty, \infty}$.

Geometric Interpretation

Let $f$ be a convex real function.

Then:

for every pair of points $A$ and $B$ on the graph of $f$, the chord $AB$ lies entirely above the graph.

Examples

Square Function

The square function:

$\forall x \in \R: \map f x = x^2$

is a convex real function.

Also known as

A convex function can also be referred to as:

a concave up function

a convex down function.

Also see

• Equivalence of Definitions of Convex Real Function

• Definition:Strictly Convex Real Function

• Definition:Concave Real Function
• Definition:Strictly Concave Real Function

• Real Function is Concave iff its Negative is Convex

• Results about convex real functions can be found here.