Brøndsted-Rockafellar theorem

An extended-real-valued function $ f $ on a Banach space $ E $ over the real numbers $ \mathbf R $ is said to be proper if $ - \infty < f ( x ) \leq \infty $ for all $ x \in E $ and $ f ( x ) < \infty $ for at least one point $ x $. The epigraph of such a function is the subset of the product space $ E \times \mathbf R $ defined by

$$ { \mathop{\rm epi} } ( f ) = \left \{ {( x, r ) \in E \times \mathbf R } : {x \in E, r \geq f ( x ) } \right \} . $$

The function $ f $ is convex (cf. Convex function (of a real variable)) precisely when the set $ { \mathop{\rm epi} } ( f ) $ is convex (cf. Convex set) and $ f $ is lower semi-continuous (cf. Semi-continuous function) precisely when $ { \mathop{\rm epi} } ( f ) $ is closed (cf. Closed set). A continuous linear functional $ x ^ {*} $ on $ E $( that is, a member of the dual space $ E ^ {*} $) is said to be a subgradient of $ f $ at the point $ x $ provided $ f ( x ) < \infty $ and $ x ^ {*} ( y - x ) \leq f ( y ) - f ( x ) $ for all $ y \in E $. The set of all subgradients to $ f $ at $ x $( where $ f ( x ) $ is finite) forms the subdifferential $ \partial f ( x ) $ of $ f $ at $ x $. The Brøndsted–Rockafellar theorem [a2] asserts that for a proper convex lower semi-continuous function $ f $, the set of points where $ \partial f ( x ) $ is non-empty is dense in the set of $ x $ where $ f $ 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 $ x $ can be identified with a support functional (cf. Support function) of $ { \mathop{\rm epi} } ( f ) $ at the point $ ( x, f ( x ) ) $. 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