Closed convex function

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 $$ f $$ is closed.

This definition is valid for any function, but most used for convex functions. A proper convex function is closed if and only if it is lower semi-continuous.

Properties

 * If $$f: \mathbb{R}^n \rightarrow \mathbb{R} $$ is a continuous function and $$\mbox{dom} f $$ is closed, then $$ f$$ is closed.
 * If $$f: \mathbb R^n \rightarrow \mathbb R $$ is a continuous function and $$\mbox{dom} f $$ is open, then $$ f $$ is closed if and only if it converges to $$\infty$$ along every sequence converging to a boundary point of $$\mbox{dom} f $$.
 * A closed proper convex function f is the pointwise supremum of the collection of all affine functions h such that h ≤ f (called the affine minorants of f).