Closed convex function

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

Equivalently, if the epigraph defined by $epi f = {(x, t) \in ℝ^{n + 1} | x \in dom f, f (x) \leq t}$ is closed, then the function $f$ 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]

Properties

If $f : ℝ^{n} \to ℝ$ is a continuous function and $dom f$ is closed, then $f$ is closed.
If $f : ℝ^{n} \to ℝ$ is a continuous function and $dom f$ is open, then $f$ is closed if and only if it converges to $\infty$ along every sequence converging to a boundary point of $dom f$ .^[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).

References

↑ Convex Optimization Theory. Athena Scientific. 2009. pp. 10, 11. ISBN 978-1886529311.
↑ Boyd, Stephen; Vandenberghe, Lieven (2004). Convex optimization (PDF). New York: Cambridge. pp. 639–640. ISBN 978-0521833783.

Rockafellar, R. Tyrrell (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. ISBN 978-0-691-01586-6.

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[1] Convex Optimization Theory. Athena Scientific. 2009. pp. 10, 11. ISBN 978-1886529311.

[2] Boyd, Stephen; Vandenberghe, Lieven (2004). Convex optimization (PDF). New York: Cambridge. pp. 639–640. ISBN 978-0521833783.

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v t e Convex analysis and variational analysis
Basic concepts	Convex combination Convex function Convex set
Topics (list)	Choquet theory Convex geometry Convex metric space Convex optimization Duality Lagrange multiplier Legendre transformation Locally convex topological vector space Simplex
Maps	Convex conjugate Concave (Closed K- Logarithmically Proper Pseudo- Quasi-) Convex function Invex function Legendre transformation Semi-continuity Subderivative
Main results (list)	Carathéodory's theorem Ekeland's variational principle Fenchel–Moreau theorem Fenchel-Young inequality Jensen's inequality Hermite–Hadamard inequality Krein–Milman theorem Mazur's lemma Shapley–Folkman lemma Robinson–Ursescu Simons Ursescu
Sets	Convex hull (Orthogonally, Pseudo-) Convex set Effective domain Epigraph Hypograph John ellipsoid Lens Radial set/Algebraic interior Zonotope
Series	Convex series related ((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (Hx), and (Hwx))
Duality	Dual system Duality gap Strong duality Weak duality
Applications and related	Convexity in economics