Brøndsted-Rockafellar theorem

An extended-real-valued function on a Banach space over the real numbers is said to be proper if for all and for at least one point . The epigraph of such a function is the subset of the product space defined by

The function is convex (cf. Convex function (of a real variable)) precisely when the set is convex (cf. Convex set) and is lower semi-continuous (cf. Semi-continuous function) precisely when is closed (cf. Closed set). A continuous linear functional on (that is, a member of the dual space ) is said to be a subgradient of at the point provided and for all . The set of all subgradients to at (where is finite) forms the subdifferential of at . The Brøndsted–Rockafellar theorem [a2] asserts that for a proper convex lower semi-continuous function , the set of points where is non-empty is dense in the set of where is finite (cf. Dense set). This is related to the Bishop–Phelps theorem [a1] (and the proof uses techniques of the latter), since a subgradient at a point can be identified with a support functional (cf. Support function) of at the point . These techniques were again applied to obtain minimization results (the Ekeland variational principle) for non-convex lower semi-continuous functions [a3]; see [a4] for a survey.

References

[a1]	E. Bishop, R.R. Phelps, "The support functionals of a convex set" P. Klee (ed.) , Convexity , Proc. Symp. Pure Math. , 7 , Amer. Math. Soc. (1963) pp. 27–35
[a2]	A. Brøndsted, R.T. Rockafellar, "On the subdifferentiability of convex functions" Proc. Amer. Math. Soc. , 16 (1965) pp. 605–611
[a3]	I. Ekeland, "On the variational principle" J. Math. Anal. Appl. , 47 (1974) pp. 324–353
[a4]	I. Ekeland, "Nonconvex minimization problems" Bull. Amer. Math. Soc. (NS) , 1 (1979) pp. 443–474