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 cannot be found elsewhere, and great beyond great pictures (which really help to understand the mathematics). Grab it!



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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