Positivity (2011) 15:527–536
Duality between Fréchet differentiability and strong
Received: 14 June 2010 / Accepted: 19 November 2010 / Published online: 4 December 2010
© Springer Basel AG 2010
Abstract This paper revisits the duality between differentiability and strict or strong
convexity under the Legendre–Fenchel transform f → f
. Functions f deﬁned on a
Banach space X are considered. For a lower semicontinuous but not necessarily con-
vex function f we relate essential Fréchet differentiability of the conjugate function
to essential strong convexity of f .
Keywords Legendre–Fenchel transform · Essential Gâteaux differentiability ·
Essential Fréchet differentiability · Essential strict convexity · Essential strong
Mathematics Subject Classiﬁcation (2000) 46G05 · 49J50 · 46N10
This paper deals with the duality between strict convexity and differentiability under
the Legendre–Fenchel transform. This topic has received attention for several decades.
We recall Rockafellar’s theorem on this subject [7, Thm. 26.3].
Theorem 1 (Rockafellar) Let f : R
→ R be a proper, lower semicontinuous and
convex function. Then the conjugate f
of f is essentially differentiable if and only if
f is essentially strictly convex.
Some basic concepts and definitions from convex analysis including the defini-
tion of the conjugate function f
are recalled in the next section. A generalization
T. Strömberg (
Department of Mathematics, Luleå University of Technology,
971 87 Luleå, Sweden