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Category Archives: Convex polytopes
Faces of Simple 4 Polytopes
In the conference celebrating Klee and Grünbaum’s mathematics at Seattle Günter Ziegler proposed the following bold conjecture about 4 polytopes. Conjecture: A simple 4polytope with facets has at most a linear number (in ) two dimensional faces which are not 4gons! If the polytope … Continue reading
Posted in Convex polytopes
3 Comments
IPAM Workshop – Efficiency of the Simplex Method: Quo vadis Hirsch conjecture?
Workshop at IPAM: January 18 – 21, 2011 Here is the link to the IPAM conference.
The Polynomial Hirsch Conjecture: The Crux of the Matter.
Consider t disjoint families of subsets of {1,2,…,n}, . Suppose that (*) For every , and every and , there is which contains . The basic question is: How large can t be??? Let’s call the answer f(n). … Continue reading
Posted in Combinatorics, Convex polytopes, Open problems, Polymath3
5 Comments
“A Counterexample to the Hirsch Conjecture,” is Now Out
Francisco (Paco) Santos’s paper “A Counterexample to the Hirsch Conjecture” is now out: For some further information and links to the media see also this page. Here is a link to a TV interview. Abstract: The Hirsch Conjecture (1957) … Continue reading
Test Your Intuition (12): Perturbing a Polytope
Let P be a ddimensional convex polytope. Can we always perturb the vertices of P moving them to points with rational coordinates without changing the combinatorial structure of P? In order words, you require that a set of vertices whose … Continue reading
Posted in Convex polytopes, Test your intuition
Tagged Convex polytopes, Test your intuition
4 Comments
Francisco Santos Disproves the Hirsch Conjecture
A title and an abstract for the conference “100 Years in Seattle: the mathematics of Klee and Grünbaum” drew a special attention: Title: “A counterexample to the Hirsch conjecture” Author: Francisco Santos, Universidad de Cantabria Abstract: I have been in … Continue reading
Posted in Convex polytopes, Open problems, Polymath3
36 Comments
Plans for polymath3
Polymath3 is planned to study the polynomial Hirsch conjecture. In order not to conflict with Tim Gowers’s next polymath project which I suppose will start around January, I propose that we will start polymath3 in mid April 2010. I plan to write a … Continue reading
Why are Planar Graphs so Exceptional
Harrison Brown asked the problem “Why are planar graphs so exceptional” over mathoverflow, and I was happy to read it since it is a problem I have often thought about over the years, as I am sure have many combinatorialsists and graph … Continue reading
Posted in Combinatorics, Convex polytopes
2 Comments
The Polynomial Hirsch Conjecture: Discussion Thread, Continued
Here is a link for the justposted paper Diameter of Polyhedra: The Limits of Abstraction by Freidrich Eisenbrand, Nicolai Hahnle, Sasha Razborov, and Thomas Rothvoss. And here is a link to the paper by Sandeep Koranne and Anand Kulkarni “The dstep Conjecture is Almost true” – … Continue reading
Posted in Convex polytopes, Open discussion, Open problems
Tagged Convex polytopes, Hirsch conjecture
16 Comments
(Eran Nevo) The gConjecture III: Algebraic Shifting
This is the third in a series of posts by Eran Nevo on the gconjecture. Eran’s first post was devoted to the combinatorics of the gconjecture and was followed by a further post by me on the origin of the gconjecture. … Continue reading
Posted in Combinatorics, Convex polytopes, Guest blogger, Open problems
Tagged gconjecture, Shifting
3 Comments