Category Archives: Convex polytopes

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

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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

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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

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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

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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

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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

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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

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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

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