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Do you know any reference where I can find some results in this sense:

Consider $W:K\to [0,\infty)$ is a functional defined on a convex cone $K\subset X$, where $X$ is a Banach space. Then the convexity of $W$ is equivalent to the monotonicity of its set-valued subdifferential $\partial W$.

I saw this assertion in this article https://www.researchgate.net/publication/321069893_A_px-version_of_Diaz-Saa_Inequality_and_some_applications (at page 3). Here it is a print-screen:

PRINT SCREEN

So, I am interested in sources that deal with functionals defined on cones (not on the entire space), and if they link subdifferentiability with Gateaux differentiability then it would be awesome.

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  • $\begingroup$ Many books on convex analysis define convex function mapping into $]-∞,∞]$ and define the effective domain of the function as the set where it has finite values. In this contact the claim is also true, so I am not really sure what the question is. $\endgroup$
    – Dirk
    Commented Aug 16 at 18:27
  • $\begingroup$ I think that the answer depends crucially on the definition of "subdifferential". If we take the standard definition $\partial W(x) = \{ u \mid \forall y \in K : W(y) \ge W(x) + \langle u, y - x \rangle\}$, then the answer is "no". Take $K = [0, \infty) \subset \mathbb R$ and $W(x) = \sqrt{x}$. Then, the subdifferential is empty everywhere (up to the origin) and, consequently, the subdifferential mapping is monotone. $\endgroup$
    – gerw
    Commented Sep 3 at 8:46

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