Closed convex function

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In mathematics, a function f: \mathbb{R}^n \rightarrow \mathbb{R} is said to be closed if for each  \alpha \in \mathbb{R}, the sublevel set  \{ x \in \mbox{dom} f \vert f(x) \leq \alpha \} is a closed set.

Equivalently, if the epigraph defined by  \mbox{epi} f = \{ (x,t) \in \mathbb{R}^{n+1} \vert x \in \mbox{dom} f,\; f(x) \leq t\} is closed, then the function  f(x) is closed.

This definition is valid for any function, but most used for convex function. A proper convex function is closed if and only if it is lower semi-continuous. For a convex function which is not proper there is disagreement as to the definition of the closure of the function.[citation needed]

Properties[edit]

  • A closed proper convex function f is the pointwise supremum of the collection of all affine functions h such that hf (called the affine minorants of f).


References[edit]