we say that is a convex function. If for any and any , we have
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 , a continuous function is convex if and only if for all , we have
Suppose is twice continuously differentiable function on . Then is convex if and only if . If , then is strictly convex.
,, and are convex functions on . Also, is strictly convex, but vanishes at .
A norm (http://planetmath.org/NormedVectorSpace) 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 be a subset of a vector space over the reals, and an extended real-valued function defined on . The epigraph of , denoted by , is the set
An extended real-valued function defined on a subset of a vector space over the reals is said to be convex if its epigraph is a convex subset of . With this definition, the domain of need not be convex. However, its subset , called the effective domain and denoted by , is convex. To see this, suppose and with . Then , where , since is convex by definition. Therefore, . In fact, , which implies that .
|Date of creation||2013-03-22 11:46:26|
|Last modified on||2013-03-22 11:46:26|
|Last modified by||matte (1858)|
|Defines||strictly convex function|
|Defines||strictly concave function|