Problems on polytopes, their groups, and realizations

Abstract

The paper gives a collection of open problems on abstract polytopes that were either presented at the Polytopes Day in Calgary or motivated by discussions at the preceding Workshop on Convex and Abstract Polytopes at the Banff International

Developments and open problems on chiral polytopes

“…In [37, Chapter 6B] a general theory on locally spherical regular polytopes is provided, and every such regular polytope is shown to be a tessellation of a spherical, Euclidean or hyperbolic space-form arising as a quotient (from a combinatorial, geometric and topological point of view) of a regular tessellation of the spherical, Euclidean or hyperbolic space by a normal sparse subgroup. Similar arguments can be applied to prove an analogous result for chiral polytopes (see also [60,Section 5]). …”

“…Whenever there exist chiral polytopes with facets isomorphic to a regular or chiral (d − 1)-polytope K 1 and vertex-figures isomorphic to a regular or chiral polytope K 2 , but not both K 1 and K 2 regular, there exists a universal polytope {K 1 , K 2 } ch with the property that every polytope with facets isomorphic to K 1 and vertex-figures isomorphic to K 2 is a quotient of {K 1 , K 2 } ch (see [60,Theorem 2]). This will be discussed in more detail in Section 7.…”

“…(A computer based method was also used to construct the atlas in [24].) Then a proof of existence of finite d-polytopes for every rank d ≥ 3 was given in [49], providing a positive answer to Problem 14 in [60].…”

“…Whenever there exists a chiral amalgamation of two polytopes K 1 and K 2 such that at least one K 1 and K 2 is chiral and the other one is regular or chiral, there exists a universal amalgamation {K 1 , K 2 } ch with the property that any other amalgamation of K 1 and K 2 is a quotient of it (see [57,Section 6]). We include here the chiral part of Problem 8 and Problem 11 of [60].…”

“…The structure of the cone is determined by the family of irreducible orthogonal representations of the automorphism group of P. No such theory has been developed for chiral realizations of regular or chiral polytopes. We include here Problem 21 of [60].…”