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. For a convex function which is not proper there is disagreement as to the definition of the closure of the function.

Properties

• If $f:\mathbb {R} ^{n}\rightarrow \mathbb {R}$ 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).

References

• Rockafellar, R. Tyrrell (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. ISBN 978-0-691-01586-6.