Closed Convex Function

In mathematics, a function $f: \mathbb^n \rightarrow \mathbb$ is said to be closed if for each $\alpha \in \mathbb$, the sublevel set
$\$
is a closed set.

Equivalently, if the epigraph defined by
$\mbox f = \$
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. For a convex function which is not proper there is disagreement as to the definition of the ''closure'' of the function.

Properties

* If $f: \mathbb^n \rightarrow \mathbb$ is a continuous function and $\mbox f$ is closed, then $f$ is closed.
* If $f: \mathbb R^n \rightarrow \mathbb R$ is a continuous function and $\mbox f$ is open, then $f$ is closed if and only if it converges to $\infty$ along every sequence converging to a boundary point of $\mbox f$.
* 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'').

References

*

Convex analysis
Types of functions

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