Kalai's squeezed 3-spheres are polytopal

In 1988, Kalai extended a construction of Billera and Lee to produce many triangulated (d-1)-spheres. In fact, in view of upper bounds on the number of simplicial d-polytopes by Goodman and Pollack, he derived that for every dimension d>=5, most of these (d-1)-spheres are not polytopal. However, for d=4, this reasoning fails. We can now show that, as already conjectured by Kalai, all of his 3-spheres are in fact polytopal. Moreover, we can now give a shorter proof of Hebble & Lee's 2000 result that the dual graphs of these 4-polytopes are Hamiltonian. Therefore, the polars of these Kalai polytopes yield another family supporting Barnette's conjecture that all simple 4-polytopes admit a Hamiltonian circuit.Comment: 11 pages, 5 figures; accepted for publication in J. Discrete & Computational Geometr

An explicit construction for neighborly centrally symmetric polytopes

We give an explicit construction, based on Hadamard matrices, for an infinite series of floor{sqrt{d}/2}-neighborly centrally symmetric d-dimensional polytopes with 4d vertices. This appears to be the best explicit version yet of a recent probabilistic result due to Linial and Novik, who proved the existence of such polytopes with a neighborliness of d/400.Comment: 9 pages, no figure

Dissections, Hom-complexes and the Cayley trick

We show that certain canonical realizations of the complexes Hom(G,H) and Hom_+(G,H) of (partial) graph homomorphisms studied by Babson and Kozlov are in fact instances of the polyhedral Cayley trick. For G a complete graph, we then characterize when a canonical projection of these complexes is itself again a complex, and exhibit several well-known objects that arise as cells or subcomplexes of such projected Hom-complexes: the dissections of a convex polygon into k-gons, Postnikov's generalized permutohedra, staircase triangulations, the complex dual to the lower faces of a cyclic polytope, and the graph of weak compositions of an integer into a fixed number of summands.Comment: 23 pages, 5 figures; improved exposition; accepted for publication in JCT

On the Monotone Upper Bound Problem

The Monotone Upper Bound Problem asks for the maximal number M(d,n) of vertices on a strictly-increasing edge-path on a simple d-polytope with n facets. More specifically, it asks whether the upper bound M(d,n)<=M_{ubt}(d,n) provided by McMullen's (1970) Upper Bound Theorem is tight, where M_{ubt}(d,n) is the number of vertices of a dual-to-cyclic d-polytope with n facets. It was recently shown that the upper bound M(d,n)<=M_{ubt}(d,n) holds with equality for small dimensions (d<=4: Pfeifle, 2003) and for small corank (n<=d+2: G\"artner et al., 2001). Here we prove that it is not tight in general: In dimension d=6 a polytope with n=9 facets can have M_{ubt}(6,9)=30 vertices, but not more than 26 <= M(6,9) <= 29 vertices can lie on a strictly-increasing edge-path. The proof involves classification results about neighborly polytopes, Kalai's (1988) concept of abstract objective functions, the Holt-Klee conditions (1998), explicit enumeration, Welzl's (2001) extended Gale diagrams, randomized generation of instances, as well as non-realizability proofs via a version of the Farkas lemma.Comment: 15 pages; 6 figure

Polygons as Sections of Higher-Dimensional Polytopes

We show that every heptagon is a section of a 3-polytope with 6 vertices. This implies that every n-gon with n≥7 can be obtained as a section of a (2+⌊n7⌋)-dimensional polytope with at most ⌈6n7⌉ vertices; and provides a geometric proof of the fact that every nonnegative n×m matrix of rank 3 has nonnegative rank not larger than ⌈6min(n,m)7⌉. This result has been independently proved, algebraically, by Shitov (J. Combin. Theory Ser. A 122, 2014)

Prodsimplicial-Neighborly Polytopes

Simultaneously generalizing both neighborly and neighborly cubical polytopes, we introduce PSN polytopes: their k-skeleton is combinatorially equivalent to that of a product of r simplices. We construct PSN polytopes by three different methods, the most versatile of which is an extension of Sanyal and Ziegler's "projecting deformed products" construction to products of arbitrary simple polytopes. For general r and k, the lowest dimension we achieve is 2k+r+1. Using topological obstructions similar to those introduced by Sanyal to bound the number of vertices of Minkowski sums, we show that this dimension is minimal if we additionally require that the PSN polytope is obtained as a projection of a polytope that is combinatorially equivalent to the product of r simplices, when the dimensions of these simplices are all large compared to k.Comment: 28 pages, 9 figures; minor correction