Talk:Subdifferential

The article says

"The subdifferential of the support function of a convex set coincides with the set itself"

This seems to disagree with corollary 25.5.3 of Rockafellar's book. Or am I mistaken?

Could you please explain a little, what is the problem? I do not have that book on my table. I also note that compactness is required. --Boris Tsirelson (talk) 23:22, 7 October 2013 (CEST)

This is a corollary 23.5.3 of Theorem 23.5 that relates the subdifferential of a convex function to its dual function. It says

Let C be a non empty closed convex set. Then for h the support function of C and each vector x, the subdifferential of h at x consists of the points (if any) where the linear function <.,x> achieves its maximum over C.

To me this does not agree with the claim about the subdifferential of the support function of a compact subset.--Edouard 14:14, 8 October 2013 (CEST)

Yes, it seems I see what happens. The phrase from our article does not make sense, since subdifferential at a point is a set (and the point is not specified). Probably it if forgotten to say "at 0". Then the statement looks believable, and conforms with the corollary quoted by you. Indeed, x=0, and the zero function achieves its maximum over C at every point of C. Does it make sense? --Boris Tsirelson (talk) 17:44, 8 October 2013 (CEST)

The statement is clearer. However I think this is not the deepest link between subdifferential of convex functions and duality in convex analysis, which is the purpose of this example. Putting emphasis on this might be a bit misleading (at least it was to me). This is personal taste, but I would rather put emphasis on something like x* is in the subdifferential of f (convex proper) at x if and only if f(x) + f*(x*) = <x,x*> where f* is the dual function of f. This is one part of the theorem I mentioned above and implies the assertion you propose (f* is an indicator function in this case).--Edouard 12:19, 9 October 2013 (CEST)

Maybe. I feel I am not an expert in this matter. If you feel you are, you are welcome to add the more general claim to the article. But probably the special case is also worth mentioning (its formulation is simpler). --Boris Tsirelson (talk) 14:55, 9 October 2013 (CEST)

Let's keep it simple. I just add the mention "at the origin" so that the statement is true.--Edouard 19:14, 9 October 2013 (CEST)