Polytopal Bier Spheres and Kantorovich–Rubinstein Polytopes of Weighted Cycles

Bier Spheres and Posets

In 1992 Thomas Bier presented a strikingly simple method to produce a huge number of simplicial (n− 2)-spheres on 2n vertices as deleted joins of a simplicial complex on n vertices with its combinatorial Alexander dual. Here we interpret his construction as giving the poset of all the intervals in a boolean algebra that “cut across an ideal.” Thus we arrive at a substantial generalization of Bi...

On Polytopal Upper Bound Spheres

Generalizing a result (the case k = 1) due to M. A. Perles, we show that any polytopal upper bound sphere of odd dimension 2k+1 belongs to the generalizedWalkup class Kk(2k + 1), i.e., all its vertex links are k-stacked spheres. This is surprising since the k-stacked spheres minimize the face-vector (among all polytopal spheres with given f0, . . . , fk−1) while the upper bound spheres maximize...

Kalai's squeezed 3-spheres are polytopal

Neighborly Cubical Polytopes and Spheres

We prove that the neighborly cubical polytopes studied by Günter M. Ziegler and the first author [14] arise as a special case of the neighborly cubical spheres constructed by Babson, Billera, and Chan [4]. By relating the two constructions we obtain an explicit description of a non-polytopal neighborly cubical sphere and, further, a new proof of the fact that the cubical equivelar surfaces of M...

Enumeration in Convex Geometries and Associated Polytopal Subdivisions of Spheres

We construct CW spheres from the lattices that arise as the closed sets of a convex closure, the meet-distributive lattices. These spheres are nearly polytopal, in the sense that their barycentric subdivisions are simplicial polytopes. The complete information on the numbers of faces and chains of faces in these spheres can be obtained from the defining lattices in a manner analogous to the rel...