# infinite dimensional polyhedra

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?

• 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. – Lev Borisov Feb 6 '14 at 19:41
• You might look at Paolo d'Alessandro, "Generalizing polyhedra to infinite dimension," 2011. PDF download link. – Joseph O'Rourke Feb 6 '14 at 19:42
• @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. – Wlodek Kuperberg Feb 6 '14 at 20:35
• 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) – Ankur Feb 7 '14 at 2:59
• @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. – Manfred Weis Feb 7 '14 at 10:07