Category Archives: Convex polytopes

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 | | 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

| Tagged , | 4 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 | | 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

The Polynomial Hirsch Conjecture – How to Improve the Upper Bounds.

I can see three main avenues toward making progress on the Polynomial Hirsch conjecture. One direction is trying to improve the upper bounds, for example,  by looking at the current proof and trying to see if it is wasteful and if so where … Continue reading

Posted in Convex polytopes, Open discussion, Open problems | Tagged , | 14 Comments

The Polynomial Hirsch Conjecture, a Proposal for Polymath3 (Cont.)

The Abstract Polynomial Hirsch Conjecture A convex polytope is the convex hull of a finite set of points in a real vector space. A polytope can be described as the intersection of a finite number of closed halfspaces. Polytopes have … Continue reading

Posted in Convex polytopes, Open discussion, Open problems | | 5 Comments

The Polynomial Hirsch Conjecture: A proposal for Polymath3

This post is continued here.  Eddie Kim and Francisco Santos have just uploaded a survey article on the Hirsch Conjecture. The Hirsch conjecture: The graph of a d-polytope with n vertices  facets has diameter at most n-d. We devoted several … Continue reading

(Eran Nevo) The g-Conjecture II: The Commutative Algebra Connection

Richard Stanley This post is authored by Eran Nevo. (It is the second in a series of five posts.) The g-conjecture: the commutative algebra connection Let be a triangulation of a -dimensional sphere. Stanley’s idea was to associate with a ring … Continue reading