I have a reference request which I hope some reader here can help me with.

I have encountered a set that has all the properties that one would expect from a polyhedral set (in the sense of finite dimensional convex analysis - an intersection of finitely many half-spaces), however in my case the number of intersecting half-spaces could be infinite. I am interested in things such as extreme points, extreme rays etc. However, the catch is that the set is itself in the space of functions. I am therefore looking for a principled generalization of the concept related to polyhedra to infinite-dimensional spaces. Could someone help me out with this?

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    $\begingroup$ If you have a finite number of half-spaces, then presumably you have a finite number of linear functions. Then you want to consider the span of these linear functions. This would give you a map from your infinite-dimensional space to a finite-dimensional space and all of the polytope business comes from the latter. So I think that you can separate the infinite-dimensional issues and the polyhedral issues. $\endgroup$ – Lev Borisov Feb 6 '14 at 19:41
  • $\begingroup$ You might look at Paolo d'Alessandro, "Generalizing polyhedra to infinite dimension," 2011. PDF download link. $\endgroup$ – Joseph O'Rourke Feb 6 '14 at 19:42
  • $\begingroup$ @LevBorisov: The map you describe is in fact a projecion onto a finite-dimensional space in a direction parallel to each of the defining half-spaces. Therefore the original, infinite-dimensional polytope is the Cartesian product of a finite-dimensional one with an infinite-dimensional space. You should write your comment in the "answer" box. $\endgroup$ – Wlodek Kuperberg Feb 6 '14 at 20:35
  • $\begingroup$ In my case, I could have infinitely many half-spaces intersecting. Can something be said about this case? (I edited my question with this clarification) $\endgroup$ – Ankur Feb 7 '14 at 2:59
  • $\begingroup$ @Ankur: I miss the restriction to a countable infinite number of intersecting half-spaces in the problem description; otherwise a sphere would also be an infinite convex polyhedron. Maybe in the title the restriction to convex polyhedra should also be mentioned. $\endgroup$ – Manfred Weis Feb 7 '14 at 10:07

Intersecting countably many half-spaces, you can get a general closed convex set (in a separable Banach space, say). For extreme points and other interesting things, I like:

Robert R. Phelps, Lectures on Choquet's Theorem, second edition, Lecture Notes in Mathematics, Vol. 1767. Springer- Verlag, 2001.


The theorem of Hahn Banach states that EVERY closed, convex subset of a normed space, say, is the intersection of an infinite family of closed half planes so that you are essentially talking about the geometry of such sets. A suitable introductory text would be the popular "Geometric Functional Analysis" by R.B. Holmes.


For a very good reference see Eckland and Turnbull, Infinite-Dimensional Optimization and Convexity.


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