Closed convex function
Equivalently, if the epigraph defined by is closed, then the function 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.
- If is a continuous function and is closed, then is closed.
- If is a continuous function and is open, then is closed if and only if it converges to along every sequence converging to a boundary point of .
- 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) . Convex Analysis. Princeton, NJ: Princeton University Press. ISBN 978-0-691-01586-6.
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