Closed convex function
In mathematics, a function is said to be closed if for each , the sublevel set is a closed set.
Equivalently, if the epigraph defined by is closed, then the function is closed.
This definition is valid for any function, but most used for
lower semi-continuous.[1] For a convex function that is not proper, there is disagreement as to the definition of the closure of the function.[citation needed
]
Properties
- 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 .[2]
- A closed proper convex function f is the pointwise affine functionsh such that h ≤ f (called the affine minorants of f).
References
- ISBN 978-1886529311.
- ISBN 978-0521833783.
- ISBN 978-0-691-01586-6.