Projected products of polygons

Abstract

Abstract.It is an open problem to characterize the cone of f -vectors of 4-dimensional convex polytopes. The question whether the "fatness" of the fvector of a 4-polytope can be arbitrarily large is a key problem in this context.Here we construct a 2-parameter family of 4-dimensional polytopes π(P 2r n ) with extreme combinatorial structure. In this family, the "fatness" of the fvector gets arbitrarily close to 9; an analogous invariant of the flag vector, the "complexity," gets arbitrarily close to 16.The polytopes are obtained from suitable deformed products of even polygons by a projection to R 4 .

Extension Complexity of Polytopes with Few Vertices or Facets

“…In order to study possible candidates for Q, we will use (a simplified version of) the Projection Lemma of Sanyal and Ziegler [25,34], which can be also understood in terms of McMullen's transforms and diagrams [18]. It characterizes the faces preserved under projections.…”

Prodsimplicial-Neighborly Polytopes

“…For any face F of P , let ϕ(F ) denote the set of indices of the facets of P containing F , i.e., such that F = i∈ϕ(F ) F i . Lemma 3.2 (Projection Lemma [AZ99,Zie04]). A face F of the polytope P is strictly preserved under the projection π if and only if {g i | i ∈ ϕ(F )} is positively spanning.…”

“…(2) If P i is an even cycle, then n i = 2, m i = 2p and χ i = 2, so that ∆ = 2p − 3. This yields projected products of polygons -see [Zie04,SZ09].…”

“…(3) Deformed Product constructions in the spirit of Sanyal & Ziegler [Zie04,SZ09]. The second part derives topological obstructions for the existence of such objects, using techniques developed by Sanyal in [San09] (see also [RS09]) to bound the number of vertices of Minkowski sums.…”

“…Finally, we extend Sanyal & Ziegler's technique of "projecting deformed products of polygons" [Zie04,SZ09] to products of arbitrary simple polytopes: given a polytope P that is combinatorially equivalent to a product of simple polytopes, we exhibit a suitable projection that preserves the complete k-skeleton of P . More concretely, we describe how to use colorings of the graphs of the polar polytopes of the factors in the product to raise the dimension of the preserved skeleton.…”