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)