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
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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 |
Brøndsted-Rockafellar theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Br%C3%B8ndsted-Rockafellar_theorem&oldid=41305