Английская Википедия:Closed convex function

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

Equivalently, if the epigraph defined by <math> \mbox{epi} f = \{ (x,t) \in \mathbb{R}^{n+1} \vert x \in \mbox{dom} f,\; f(x) \leq t\} </math> is closed, then the function <math> f </math> 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 that is not proper, there is disagreement as to the definition of the closure of the function.Шаблон:Citation needed

Properties

• If <math>f: \mathbb{R}^n \rightarrow \mathbb{R} </math> is a continuous function and <math>\mbox{dom} f </math> is closed, then <math> f</math> is closed.
• If <math>f: \mathbb R^n \rightarrow \mathbb R </math> is a continuous function and <math>\mbox{dom} f </math> is open, then <math> f </math> is closed if and only if it converges to <math>\infty</math> along every sequence converging to a boundary point of <math>\mbox{dom} f </math>.
• 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

• Шаблон:Cite book