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Five Open Problems Regarding Convex Polytopes
This entry was posted in Combinatorics, Convex polytopes, Convexity, Open problems and tagged Combinatorics: geometric, Convex polytopes, Convexity, mathematical open problems. Bookmark the permalink.
Welcome to the blogosphere! I found this post through Google blog search but, now that I know about it, will check here more regularly.
I posted a followup here. I’m especially curious about the 5simple 5simplicial question. Infinitely many 2simplicial 2simple polytopes are known (see the EKZ paper and arXiv:math.MG/0304492) but is the same true for 3simplicial 3simple?
Dear David, thanks! I am not sure if infinitly many 3simplicial 3simple dpolytopes are known for d=6 or some other fixed d. It was also conjectured that 2simplicial 2simple 4polytopes are dense in the space of all convex body in . But this is also unknown.
What about Hirsch’s conjecture?
Dear Shashi, the Hirsch’s conjecture deseves a special post; unfortunately not much progress was made for a long time.
Just a little preview – The Hirsch Conjecture asserts that the diameter of a graph of a dpolytope with n facets is at most nd. It is not even known if there is a polynomial upper bound (in n and d ).
Answering a question from email: the construction for the number of intermediate faces of a dpolytope with O(n) vertices and facets is to embed a collection of ngons on mutually skew 2planes and take their convex hull. The resulting polytope has an fvector that’s the convolution of the fvectors of the polygons. Thus, one polygon in two dimensions has the fvector (1,n,n,1); two of them on skew planes in have the 5vector , etc. In general from fvectors and you get a convolved vector .
I made this observation in an email to Jeff Erickson and Nina Amenta in July 1997, but I think the only place it has been publically described is Jeff’s web page http://compgeom.cs.uiuc.edu/~jeffe/open/intricate.html
that should be n^{floor((d+1)/3)} and Σ_{a+b=i} A[a]B[b]. I guess wordpress strips out sub and sup html markup in comments?
Dear David,
I tried to edit your comment. To write latex here one has to do
dollarsign latex formula dollarsign
A common mistake for me is to write say $\latex \bar \Xi e^x$ this will not work because of the \ before latex, now I will delete it and try again , voila!!
Regarding polytopes, you raised the problem “what is the maximum number of kfaces for a dpolytope with n vertices and n facets.” It is quite interesting! (Barany’s question ask for the minimum number.)
Thanks! By the way, where did you get the photo of Günter? Under what conditions is it usable? I like it better than the boring snapshot that’s on his Wikipedia article now.
Thanks for your reply – I am looking forward to your post on Hirsch’s conjecture.
David, Gunter is “in charge” of 2008 mathematics year in Germany and you can find many photos, interviews, and even video clips on his homepage.
wait; a polytope is simple, if it is simplicial *and*
its dual is simplicial (only saw the last part, stated explicitly
No no, A polytope is simple if its dual is simplicial. (Only the simplex is both simple and simplicial.)
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I have a linear programming problem where the polytope is described as a convex hull of an exponential number ($2^d$) of points in ddimensional space. What could be an approach? Do you first identify the set of vertices from the $2^d$ points and check which vertex maximizes the objective? Is there an efficient algorithm to identify the set of vertices from the given set of $2^d$ points?
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