PlanetMath: convex function

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convex function

(Definition)

Definition Suppose $\Omega$ is a convex set in a vector space over $\mathbb{R}$ (or $\mathbb{C}$ ), and suppose is a function $f:\Omega\to \mathbb{R}$ . If for any $x,y\in \Omega$ , $x\neq y$ and any $\lambda \in (0,1)$ , we have

$\displaystyle f\Big( \lambda x + (1-\lambda)y\Big)\leq \lambda f(x)+(1-\lambda)f(y),$

we say that

is a convex function. If for any $x,y\in \Omega$ and any $\lambda \in (0,1)$ , we have

$\displaystyle f\Big( \lambda x + (1-\lambda)y\Big)\geq \lambda f(x)+(1-\lambda)f(y),$

we say that

is a concave function. If either of the inequalities are strict, then we say that

is a strictly convex function, or a strictly concave function, respectively.

Properties

A function is a (strictly) convex function if and only if is a (strictly) concave function. For this reason, most of the below discussion only focuses on convex functions. Analogous result holds for concave functions.
On $\mathbb{R}$ , a continuous function is convex if and only if for all $x,y\in \mathbb{R}$ , we have
$\displaystyle f\left(\frac{x+y}{2}\right)\le\frac{f(x)+f(y)}{2}.$
On $\mathbb{R}$ , a once differentiable function is convex if and only if is monotone increasing.
Suppose is twice continuously differentiable function on $\mathbb{R}$ . Then is convex if and only if $f'' \ge 0$ . If , then is strictly convex.
A local minimum of a convex function is a global minimum. See this page.

Examples

, $e^{-x}$ , and are convex functions on $\mathbb{R}$ . Also, is strictly convex, but vanishes at .
A norm is a convex function.

Remark.

We may generalize the above definition of a convex function to an that of an extended real-valued function whose domain is not necessarily a convex set. First, we define what an epigraph of a function is.

Let $\Omega$ be a subset of a vector space over the reals, and an extended real-valued function defined on $\Omega$ . The epigraph of , denoted by $\operatorname{epi}(f)$ , is the set

$\displaystyle \lbrace (x,r)\mid x\in \Omega,$ $\displaystyle r\ge f(x)\rbrace.$

An extended real-valued function

defined on a subset $\Omega$ of a vector space

over the reals is said to be convex if its epigraph is a convex subset of $V\times\mathbb{R}$ . With this definition, the domain $\Omega$ of

need not be convex. However, its subset $\lbrace x\in \Omega\mid f(x) < \infty\rbrace$ , called the effective domain and denoted by $\operatorname{eff.dom}(f)$ , is convex. To see this, suppose $x,y\in \operatorname{eff.dom}(f)$ and $z=\lambda x+(1-\lambda) y$ with $0\le \lambda \le 0$ . Then $(z,\overline{z})=\lambda(x,f(x))+(1-\lambda)(y,f(y))\in \operatorname{epi}(f)$ , where $\overline{z}=\lambda f(x)+(1-\lambda)f(y)$ , since $\operatorname{epi}(f)$ is convex by definition. Therefore, $z\in \operatorname{dom}(f)$ . In fact, $f(z)\le \overline{z}=\lambda f(x)+(1-\lambda)f(y)<\infty$ , which implies that $z\in \operatorname{eff.dom}(f)$ .

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"convex function" is owned by matte. [ full author list (5) | owner history (1) ]

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Also defines:

concave function, strictly convex function, strictly concave function, strictly convex, strictly concave, epigraph, effective domain, concave

Attachments:: local minimum of convex function is necessarily global (Theorem) by stevecheng
continuity of convex functions (Result) by pbruin
convex functions lie above their supporting lines (Result) by Andrea Ambrosio
proof of criterion for convexity (Proof) by rspuzio
inflexion point (Definition) by pahio

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Cross-references: implies, convex subset, reals, subset, domain, vanishes, global minimum, local minimum, monotone increasing, differentiable function, continuous function, strictly, strict, inequalities, function, vector space, convex set
There are 27 references to this entry.

This is version 23 of convex function, born on 2001-10-15, modified 2007-08-01.
Object id is 231, canonical name is ConvexFunction.
Accessed 40953 times total.

Classification:

AMS MSC:	26B25 (Real functions :: Functions of several variables :: Convexity, generalizations)
	26A51 (Real functions :: Functions of one variable :: Convexity, generalizations)
	52A41 (Convex and discrete geometry :: General convexity :: Convex functions and convex programs)

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