Closed convex function

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

Equivalently, if the epigraph defined by ${\displaystyle {\mbox{epi}}f=\{(x,t)\in \mathbb {R} ^{n+1}\vert x\in {\mbox{dom}}f,\;f(x)\leq t\}}${\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 ${\displaystyle f}${\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 ${\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} }${\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} } is a continuous function and ${\displaystyle {\mbox{dom}}f}${\displaystyle {\mbox{dom}}f} is closed, then ${\displaystyle f}${\displaystyle f} is closed.
• If ${\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} }${\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} } is a continuous function and ${\displaystyle {\mbox{dom}}f}${\displaystyle {\mbox{dom}}f} is open, then ${\displaystyle f}${\displaystyle f} is closed if and only if it converges to ${\displaystyle \infty }${\displaystyle \infty } along every sequence converging to a boundary point of ${\displaystyle {\mbox{dom}}f}${\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.

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