Closed convex function

Short description: Terms in Maths

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

Equivalently, if the epigraph defined by $\text{epi}f=\{(x,t)\in {\mathbb{ℝ}}^{n+1}|x\in \text{dom}f,f(x)\le t\}$ 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 that is not proper, there is disagreement as to the definition of the closure of the function.

Properties

• If $f:{\mathbb{ℝ}}^n\to \mathbb{ℝ}$ is a continuous function and $\text{dom}f$ is closed, then $f$ is closed.
• If $f:{\mathbb{ℝ}}^n\to \mathbb{ℝ}$ is a continuous function and $\text{dom}f$ is open, then $f$ is closed if and only if it converges to $\mathrm{\infty }$ along every sequence converging to a boundary point of $\text{dom}f$.
• 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). Convex Analysis. Princeton, NJ: Princeton University Press. ISBN 978-0-691-01586-6.