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Category Archives: Convex polytopes
Many triangulated three-spheres!
The news Eran Nevo and Stedman Wilson have constructed triangulations with n vertices of the 3-dimensional sphere! This settled an old problem which stood open for several decades. Here is a link to their paper How many n-vertex triangulations does the 3 … Continue reading
Posted in Combinatorics, Convex polytopes, Geometry, Open problems
Tagged Eran Nevo, Stedman Wilson
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Richard Stanley: How the Proof of the Upper Bound Theorem (for spheres) was Found
The upper bound theorem asserts that among all d-dimensional polytopes with n vertices, the cyclic polytope maximizes the number of facets (and k-faces for every k). It was proved by McMullen for polytopes in 1970, and by Stanley for general triangulations … Continue reading
Lionel Pournin found a combinatorial proof for Sleator-Tarjan-Thurston diameter result
I just saw in Claire Mathieu’s blog “A CS professor blog” that a simple proof of the Sleator-Tarjan-Thurston’s diameter result for the graph of the associahedron was found by Lionel Pournin! Here are slides of his lecture “The diameters of associahedra” … Continue reading
Posted in Combinatorics, Computer Science and Optimization, Convex polytopes
Tagged Associahedron, Lionel Pournin
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Karim Adiprasito: Flag simplicial complexes and the non-revisiting path conjecture
This post is authored by Karim Adiprasito The past months have seen some exciting progress on diameter bounds for polytopes and polytopal complexes, both in the negative and in the positive direction. Jesus de Loera and Steve Klee described simplicial polytopes which are not … Continue reading
Posted in Convex polytopes, Guest blogger
Tagged Convex polytopes, Flag complexes, Hirsch conjecture, Karim Adiprasito
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Tokyo, Kyoto, and Nagoya
Near Nagoya: Firework festival; Kyoto: with Gunter Ziegler; with Takayuki Hibi, Hibi, Marge Bayer, Curtis Green and Richard Stanly; Tokyo: Peter Frankl; crowded crossing I just returned from a trip to Japan to the FPSAC 2012 at Nagoya and a … Continue reading
Posted in Combinatorics, Conferences, Convex polytopes
Tagged Alternating sign matrices, Convex polytopes, FPSAC, Japan
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Satoshi Murai and Eran Nevo proved the Generalized Lower Bound Conjecture.
Satoshi Murai and Eran Nevo have just proved the 1971 generalized lower bound conjecture of McMullen and Walkup, in their paper On the generalized lower bound conjecture for polytopes and spheres . Let me tell you a little about it. … Continue reading
Updates, Boolean Functions Conference, and a Surprising Application to Polytope Theory
The Debate continues The debate between Aram Harrow and me on Godel Lost letter and P=NP (GLL) regarding quantum fault tolerance continues. The first post entitled Perpetual motions of the 21th century featured mainly my work, with a short response by Aram. … Continue reading
Projections to the TSP Polytope
Michael Ben Or told me about the following great paper Linear vs. Semidefinite Extended Formulations: Exponential Separation and Strong Lower Bounds by Samuel Fiorini, Serge Massar, Sebastian Pokutta, Hans Raj Tiwary and Ronald de Wolf. The paper solves an old conjecture … Continue reading
Polymath3 (PHC6): The Polynomial Hirsch Conjecture – A Topological Approach
This is a new polymath3 research thread. Our aim is to tackle the polynomial Hirsch conjecture which asserts that there is a polynomial upper bound for the diameter of graphs of -dimensional polytopes with facets. Our research so far was … Continue reading
Posted in Convex polytopes, Geometry, Polymath3
Tagged Hirsch conjecture, Polymath3, Topological combinatorics
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