convex function

Definition Suppose Ω is a convex set in a vector space over (or ), and suppose f is a function f:Ω. If for any x,yΩ, xy and any λ(0,1), we have


we say that f is a convex function. If for any x,yΩ and any λ(0,1), we have


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


  • 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 , a continuous functionMathworldPlanetmath is convex if and only if for all x,y, we have

  • On , a once differentiable function is convex if and only if f is monotone increasing.

  • Suppose f is twice continuously differentiable function on . Then f is convex if and only if f′′0. If f′′>0, then f is strictly convex.

  • A local minimumMathworldPlanetmath of a convex function is a global minimum. See this page (


  • ex,e-x, and x2 are convex functions on . Also, x4 is strictly convex, but 12x2 vanishes at x=0.

  • A norm ( is a convex function.


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 Ω be a subset of a vector space over the reals, and f an extended real-valued function defined on Ω. The epigraph of f, denoted by epi(f), is the set

{(x,r)xΩ, rf(x)}.

An extended real-valued function f defined on a subset Ω of a vector space V over the reals is said to be convex if its epigraph is a convex subset of V×. With this definition, the domain Ω of f need not be convex. However, its subset {xΩf(x)<}, called the effective domain and denoted by eff.dom(f), is convex. To see this, suppose x,yeff.dom(f) and z=λx+(1-λ)y with 0λ0. Then (z,z¯)=λ(x,f(x))+(1-λ)(y,f(y))epi(f), where z¯=λf(x)+(1-λ)f(y), since epi(f) is convex by definition. Therefore, zdom(f). In fact, f(z)z¯=λf(x)+(1-λ)f(y)<, which implies that zeff.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