convex function
Definition Suppose $\mathrm{\Omega}$ is a convex set in a vector space over $\mathbb{R}$ (or $\u2102$), and suppose $f$ is a function $f:\mathrm{\Omega}\to \mathbb{R}$. If for any $x,y\in \mathrm{\Omega}$, $x\ne y$ and any $\lambda \in (0,1)$, we have
$$f\left(\lambda x+(1\lambda )y\right)\le \lambda f(x)+(1\lambda )f(y),$$ 
we say that $f$ is a convex function. If for any $x,y\in \mathrm{\Omega}$ and any $\lambda \in (0,1)$, we have
$$f\left(\lambda x+(1\lambda )y\right)\ge \lambda f(x)+(1\lambda )f(y),$$ 
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

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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.

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On $\mathbb{R}$, a continuous function^{} is convex if and only if for all $x,y\in \mathbb{R}$, we have
$$f\left(\frac{x+y}{2}\right)\le \frac{f(x)+f(y)}{2}.$$ 
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On $\mathbb{R}$, a once differentiable function is convex if and only if ${f}^{\prime}$ is monotone increasing.

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

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

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

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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 realvalued function whose domain is not necessarily a convex set. First, we define what an epigraph of a function is.
Let $\mathrm{\Omega}$ be a subset of a vector space over the reals, and $f$ an extended realvalued function defined on $\mathrm{\Omega}$. The epigraph of $f$, denoted by $\mathrm{epi}(f)$, is the set
$$\{(x,r)\mid x\in \mathrm{\Omega},\text{}r\ge f(x)\}.$$ 
An extended realvalued function $f$ defined on a subset $\mathrm{\Omega}$ 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 $\mathrm{\Omega}$ of $f$ need not be convex. However, its subset $$, called the effective domain and denoted by $\mathrm{eff}.\mathrm{dom}(f)$, is convex. To see this, suppose $x,y\in \mathrm{eff}.\mathrm{dom}(f)$ and $z=\lambda x+(1\lambda )y$ with $0\le \lambda \le 0$. Then $(z,\overline{z})=\lambda (x,f(x))+(1\lambda )(y,f(y))\in \mathrm{epi}(f)$, where $\overline{z}=\lambda f(x)+(1\lambda )f(y)$, since $\mathrm{epi}(f)$ is convex by definition. Therefore, $z\in \mathrm{dom}(f)$. In fact, $$, which implies that $z\in \mathrm{eff}.\mathrm{dom}(f)$.
Title  convex function 
Canonical name  ConvexFunction 
Date of creation  20130322 11:46:26 
Last modified on  20130322 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 1801 
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 