Closed convex function

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

Equivalently, if the epigraph defined by epi f = { ( x , t ) ∈ R n + 1 | x ∈ dom f , f ( x ) ≤ t } {\displaystyle {\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 {\displaystyle 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.^[1]

• If f : R n → R {\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} } is a continuous function and dom f {\displaystyle {\mbox{dom}}f} is closed, then f {\displaystyle f} is closed.
• If f : R n → R {\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} } is a continuous function and dom f {\displaystyle {\mbox{dom}}f} is open, then f {\displaystyle f} is closed if and only if it converges to ∞ {\displaystyle \infty } along every sequence converging to a boundary point of dom f {\displaystyle {\mbox{dom}}f} .^[2]
• 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).

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