Helly’s Theorem, “Hypertrees”, and Strange Enumeration I

1. Helly’s theorem and Cayley’s formula

Helly’s theorem asserts: For a family of n convex sets in R^d, n > d, if every d+1 sets in the family have a point in common then all members in the family have a point in common.

Cayley’s formula asserts: The number of  trees on n labelled vertices is n ^{n-2}.

In this post (in two parts) we will see how an extension of Helly’s theorem has led to high dimensional analogs of Cayley’s theorem. 

  

left: Helly’s theorem demonstrated in the Stanford Encyclopedia of Philosophy (!), right: a tree 

2. Background

This post is based on my lecture at Marburg. The conference there was a celebration of new doctoral theses Continue reading

Five Open Problems Regarding Convex Polytopes

  

The problems 

1. The 3^d conjecture

A centrally symmetric d-polytope has at least 3^d non empty faces.

2. The cube-simplex conjecture

For every k there is f(k) so that every d-polytope with d \ge f(k) has a k-dimensional face which is either a simplex or combinatorially isomorphic to a k-dimsnional cube.

3. Barany’s question

For every d-dimensional polytope P and every k, 0 \le k \le d-1,  is it true that f_k(P) \ge \min (f_0(P),f_{d-1}(P))?

(In words: the number of k-dimensional faces of P is at least the minimum between the number of vertices of P and the number of facets of P. )

4.  Fat 4-polytopes

For 4-polytopes P, is the quantity (f_1(P)+f_2(P))/(f_0(P)+f_3(P)) bounded from above by some absolute constant? 

5.  five-simplicial five-simple polytopes

Are there 5-simplicial 5-simple 10-polytopes? Or at least 5-simplicial 5-simple d-polytope for some d?

(A polytope P is k-simplicial if all its faces of dimension at most k, are simplices. A polytope P is k-simple if its dual P* is k-simplicial.)

And on a personal note: My beloved, beautiful,  and troubled country celebrates 60 today: happy birthday! 

Update (May 12): David Eppstein raised in a followup a sort of a dual question to Barany’s. For a d-polytope with n vertices and n facets what is the maximal number of k-faces. For a fixed d and large n the free join of pentagons is conjectured to give asymptotically the best upper bound.

Update (July 29) Gunter Ziegler reminded me of the following additional problem of Barany: Is the number of saturated chains in a d-polytope bounded by some constant (depending on d) times the total number of faces (of all dimensions) of the polytope. A saturated flag is a 0-face inside a 1-face inside a 2-face … inside a (d-1)-face. 

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