Speaker: Daniel Kalmanovich (HUJI)
Title: Cubical polytopes
Abstract:
Convex polytopes have fascinated people for ages, and questions about their combinatorics and their geometry have been widely studied.
A fundamental goal is to understand the possible face numbers of d-dimensional polytopes. While an answer to this question is not known for any d\ge 4, if we restrict our attention to the family of simplicial d-polytopes, the celebrated g-theorem gives a complete answer. We address this question for the family of cubical polytopes. We construct sequences of cubical d-polytopes that approach the extremal rays of Adin's cone, the cone conjectured by Adin and Babson--Billera--Chan to contain the f-vectors of all cubical d-polytopes. Our polytopes also demonstrate that a natural cubical analog of the simplicial Generalized Lower Bound Theorem does not hold.
Another question, one of geometric flavor, is to understand the space of all possible realizations of a given (combinatorial type of a) polytope. We show that the realization space of the d-cube is contractible. We use this fact to define an analog of the connected sum construction for cubical polytopes, and apply this construction to the above-mentioned `extremal sequences' to conclude that the rays spanned by f-vectors of cubical d-polytopes are dense in Adin's cone. Furthermore, our contractibility result extends to the realization space of any product of simplices.
This talk is based on joint works with Ron Adin, Karim Adiprasito and Eran Nevo.