It is well known that the convex conjugate of a function that is defined as follows is continuous, However, when is it differentiable?
F*(y)=sup_{x} {x.y - F(x) }
you need F differentiable and F' one to one, which is the case if F is strictly convex (then there is a unique minimizer for the definition of F*).
but can you bring up a good reference that support your claim ?
what is the condition on the domain of F that is a set valued function of y?
F(x,y): G \arrow R^N
F*(y)=sup_{ x \in H(x)} {x.y - F(x)
where H is a set valued function (or correspondence) that is equivalent to
H: D \twoheadrightarrow R^M
(x,y) \in G(x,y) \equiv x \in H(y)
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Ari-Pekka Perkkiö
Ludwig-Maximilians-Universität München
Check Rockafellar's Convex Analysis (Theorem 26.1 and Theorem 26.3). The relevant concepts are "essential strict convexity" and "essential smoothness"
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Behrad Mahboobi
Science and Research Branch, Islamiz Azad University
@ Ari-Pekka Perkkiö , thank you very much, is any theorem that guarantee quasi convexity or quasi concavity of the argument of convex conjugate :
x(x*)=argsup_{ x \in H(x)} {x.x* - F(x)}
when x(x*) is convex versus x*
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