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- To cheer you up in difficult times 32, Annika Heckel's guest post: How does the Chromatic Number of a Random Graph Vary?
- Amazing: Karim Adiprasito proved the g-conjecture for spheres!
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# Tag Archives: Convex polytopes

## Karim Adiprasito: Flag simplicial complexes and the non-revisiting path conjecture (A combinatorial proof of the Adiprasito-Benedetti theorem.)

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. Added later: Mazi and I at the same restaurant taken by Stanley. I just returned from … Continue reading

Posted in Combinatorics, Conferences, Convex polytopes
Tagged Alternating sign matrices, Convex polytopes, FPSAC, Japan
2 Comments

## 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

## Test Your Intuition (12): Perturbing a Polytope

Let P be a d-dimensional 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

## The Polynomial Hirsch Conjecture: Discussion Thread, Continued

Here is a link for the just-posted 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 d-step Conjecture is Almost true” – … Continue reading

Posted in Convex polytopes, Open discussion, Open problems
Tagged Convex polytopes, Hirsch conjecture
16 Comments

## Igor Pak’s “Lectures on Discrete and Polyhedral Geometry”

Here is a link to Igor Pak’s book on Discrete and Polyhedral Geometry (free download) . And here is just the table of contents. It is a wonderful book, full of gems, contains original look on many important directions, things that … Continue reading

Posted in Book review, Convex polytopes, Convexity
Tagged Convex polytopes, Convexity, Igor Pak, rigidity
4 Comments

## Five Open Problems Regarding Convex Polytopes

The problems 1. The conjecture A centrally symmetric d-polytope has at least non empty faces. 2. The cube-simplex conjecture For every k there is f(k) so that every d-polytope with has a k-dimensional face which is either a simplex … Continue reading