Category Archives: Convex polytopes

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

Posted in Convex polytopes | Tagged , | 2 Comments

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 , | 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 counter-example to the Hirsch conjecture” Author: Francisco Santos, Universidad de Cantabria Abstract:  I have been in … Continue reading

Posted in Convex polytopes, Open problems, Polymath3 | 34 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

Posted in Convex polytopes, Polymath3 | Tagged , | 4 Comments

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 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 , | 16 Comments

(Eran Nevo) The g-Conjecture III: Algebraic Shifting

This is the third in a series of posts by Eran Nevo on the g-conjecture. Eran’s first post was devoted to the combinatorics of the g-conjecture and was followed by a further post by me on the origin of the g-conjecture. … Continue reading

Posted in Combinatorics, Convex polytopes, Guest blogger, Open problems | Tagged , | 2 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 , , , | 4 Comments

The Polynomial Hirsch Conjecture: Discussion Thread

This post is devoted to the polymath-proposal about the polynomial Hirsch conjecture. My intention is to start here a discussion thread on the problem and related problems. (Perhaps identifying further interesting related problems and research directions.) Earlier posts are: The polynomial Hirsch … Continue reading

Posted in Convex polytopes, Open discussion, Open problems | Tagged , | 115 Comments