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# Category Archives: Convex polytopes

## A Diameter problem (7): The Best Known Bound

Our Diameter problem for families of sets Consider a family of subsets of size d of the set N={1,2,…,n}. Associate to a graph as follows: The vertices of are simply the sets in . Two vertices and are adjacent if . … Continue reading

## A Diameter Problem (6): Abstract Objective Functions

George Dantzig and Leonid Khachyan In this part we will not progress on the diameter problem that we discussed in the earlier posts but will rather describe a closely related problem for directed graphs associated with ordered families of sets. The role models for … Continue reading

Posted in Combinatorics, Convex polytopes, Open problems
Tagged Hirsch conjecture, Linear programming
7 Comments

## A Diameter Problem (5)

6. First subexponential bounds. Proposition 1: How to prove it: This is easy to prove: Given two sets and in our family , we first find a path of the form where, and . We let with and consider the family … Continue reading

## Diameter Problem (4)

Let us consider another strategy to deal with our diameter problem. Let us try to associate other graphs to our family of sets. Recall that we consider a family of subsets of size of the set . Let us now associate … Continue reading

## Diameter Problem (3)

3. What we will do in this post and and in future posts We will now try all sorts of ideas to give good upper bounds for the abstract diameter problem that we described. As we explained, such bounds apply … Continue reading

Posted in Combinatorics, Convex polytopes, Open problems
Tagged Hirsch conjecture, Linear programming, Quasi-automated proofs
1 Comment

## A Diameter Problem (2)

2. The connection with Hirsch’s Conjecture The Hirsch Conjecture asserts that the diameter of the graph G(P) of a d-polytope P with n facets is at most n-d. Not even a polynomial upper bound for the diameter in terms of d and … Continue reading

Posted in Combinatorics, Convex polytopes, Open problems
5 Comments

## A Diamater Problem for Families of Sets.

Let me draw your attention to the following problem: Consider a family of subsets of size d of the set N={1,2,…,n}. Associate to a graph as follows: The vertices of are simply the sets in . Two vertices and are adjacent … Continue reading

Posted in Combinatorics, Convex polytopes, Open problems
10 Comments

## Euler’s Formula, Fibonacci, the Bayer-Billera Theorem, and Fine’s CD-index

Bill Gessley proving Euler’s formula (at UMKC) In the earlier post about Billerafest I mentioned the theorem of Bayer and Billera on flag numbers of polytopes. Let me say a little more about it. 1. Euler Euler’s theorem … Continue reading

Posted in Combinatorics, Convex polytopes
Tagged Bayer-Billera's theorem, CD-index, Flag numbers
3 Comments

## Billerafest

I am unable to attend the conference taking place now at Cornell, but I send my warmest greetings to Lou from Jerusalem. The titles and abstracts of the lectures can be found here. Let me tell you about two theorems by Lou. … Continue reading

Posted in Conferences, Convex polytopes
Tagged f-vectors, flag vectors, g-conjecture, Lou Billera
1 Comment

## 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