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 convex functions. A proper convex function is closed if and only if it is lower semi-continuous.

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 .
• 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