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# Category Archives: Convex polytopes

## 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 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

## 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

## 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 Convex polytopes, Hirsch conjecture
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 g-conjecture, Shifting
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 Convex polytopes, Convexity, Igor Pak, rigidity
4 Comments