Talk:Dual cone and polar cone

the dual cone points in the wrong direction --unsigned
 * Well, the picture shows the polar cone, not the dual cone. I think it is right. Oleg Alexandrov (talk) 15:25, 29 May 2007 (UTC)

Merge with Convex cone?
I propose to merge this article into a section of Convex cone. At the moment, the corresponding section there is just the definition of the dual cone, without examples, properties, or illustrations, all which, I believe, should have been present there. On the other hand, this article lists the properties and has nice illustrations, but does not provide any context, and I have a hard time seeing how it can evolve into a stand-alone article. Thus the merger would be beneficial for both sides.

In the summary of his edit, Oleg commented that this is a different notion; however, I don't presently see much discussion pertaining to general sets C, besides the comment on C** for general set C. While the properties may apply to general C, it's unclear what purpose does it serve, whether such applications are common enough to warrant a separate article.

There is definitely some potential for a general article on duality in Convex geometry, which will include, for example, polarity for polytopes (cf. Polar set, another stub-orphan), but I think that such article should be allowed to grow independently, with definitions imported if necessary, of course. This is not my area of expertise, so I will not be able to write an article on duality, but I can elaborate on my other comments if needed. Arcfrk 03:45, 7 June 2007 (UTC)
 * I disagree with a merger. Some information from this article can be copied to expand the corresponding section in convex cone, but overall a merger would do no good. This article needs expansion, since it is an important concept of its own. I plan to expand it. But even now it is too big to fit into convex cone which is large by itself already. Oleg Alexandrov (talk) 04:13, 7 June 2007 (UTC)


 * As a matter of procedure, should there not also be a merge tag put into the other article, that is Convex cone? The editors of that article should also have a say in the matter. JRSpriggs 07:53, 7 June 2007 (UTC)

Mistake
Zfeinst introduced an incorrect (wrt to the rest of the article) definition of polar cone. He has copied over the defition of polar body, which is not a cone in general.
 * You are right. My bad. Zfeinst (talk) 16:34, 26 March 2012 (UTC)

Infinite dimensional situation needs attention
I edited the article to correct some type errors, since many parts of the article were written as if the dual cone belongs to the same vector space as the original cone. This is true according to some definitions of dual cone, but not according to the one with which the article begins. So I added the alternative definition. The conditions (i) and (ii) for belonging to the dual cone are cited from a book (Boyd and Vandenberghe) that I believe uses the assumption of finite dimensionality, a fixed inner product, and dual cone in the primal vector space (the latter is clear from (ii)), so I have edited the article accordingly. I don't own the book, though, so if someone wants to check whether or not they assume finite dimensionality, that would be good. They could be easily generalized to apply to the more general situation, but I'm not sure that all of the statements made below (i) and (ii) are obviously true in infinite dimension. If someone familiar with the infinite-dimensional case wants to give this some attention, that could be useful; I have mainly tried to make sure nothing false was implied. I may come back to the infinite-dimensional issue but will need to check some things first. A good reference that is careful about the infinite-dimensional case is one of the background chapters of "Lie groups, convex cones, and semigroups / Joachim Hilgert, Karl Heinrich Hofmann, and Jimmie D. Lawson, Oxford, 1989 (out of print and expensive/ hard to find, though). MorphismOfDoom (talk) 17:29, 4 June 2012 (UTC)