# convex function

Definition Suppose $\Omega$ is a convex set in a vector space over $\mathbb{R}$ (or $\mathbb{C}$), and suppose $f$ 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

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

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

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

we say that $f$ is a concave function. If either of the inequalities are strict, then we say that $f$ is a strictly convex function, or a strictly concave function, respectively.

## Properties

• A function $f$ is a (strictly) convex function if and only if $-f$ 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

 $f\left(\frac{x+y}{2}\right)\leq\frac{f(x)+f(y)}{2}.$
• On $\mathbb{R}$, a once differentiable function is convex if and only if $f^{\prime}$ is monotone increasing.

• Suppose $f$ is twice continuously differentiable function on $\mathbb{R}$. Then $f$ is convex if and only if $f^{\prime\prime}\geq 0$. If $f^{\prime\prime}>0$, then $f$ is strictly convex.

• A local minimum of a convex function is a global minimum. See this page (http://planetmath.org/ExtremalValueOfConvexconcaveFunctions).

## Examples

• $e^{x}$,$e^{-x}$, and $x^{2}$ are convex functions on $\mathbb{R}$. Also, $x^{4}$ is strictly convex, but $12x^{2}$ vanishes at $x=0$.

• A norm (http://planetmath.org/NormedVectorSpace) 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 $f$ an extended real-valued function defined on $\Omega$. The epigraph of $f$, denoted by $\operatorname{epi}(f)$, is the set

 $\{(x,r)\mid x\in\Omega,\mbox{ }r\geq f(x)\}.$

An extended real-valued function $f$ defined on a subset $\Omega$ of a vector space $V$ 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 $f$ need not be convex. However, its subset $\{x\in\Omega\mid f(x)<\infty\}$, 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 $0\leq\lambda\leq 0$. 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, $f(z)\leq\overline{z}=\lambda f(x)+(1-\lambda)f(y)<\infty$, which implies that $z\in\operatorname{eff.dom}(f)$.

 Title convex function Canonical name ConvexFunction Date of creation 2013-03-22 11:46:26 Last modified on 2013-03-22 11:46:26 Owner matte (1858) Last modified by matte (1858) Numerical id 28 Author matte (1858) Entry type Definition Classification msc 52A41 Classification msc 26A51 Classification msc 26B25 Classification msc 55Q05 Classification msc 18G30 Classification msc 18B40 Classification msc 20J05 Classification msc 20E07 Classification msc 18-01 Classification msc 20L05 Related topic JensensInequality Related topic LogarithmicallyConvexFunction Defines concave function Defines strictly convex function Defines strictly concave function Defines strictly convex Defines strictly concave Defines epigraph Defines effective domain Defines concave