# 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
How to Cite This Entry:
Brøndsted-Rockafellar theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Br%C3%B8ndsted-Rockafellar_theorem&oldid=46169
This article was adapted from an original article by R. Phelps (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article