Blending simple polytopes at faces Original Research Article

Blending two simple polytopes together at vertices, at edges, or at other supplementary faces, produces another simple polytope. In pursuing the Hirsch conjectures, vertex-blends produced polytopes which have large diameter and few diametral paths. Here we define and explore blends at higher-dimensional faces. We identify specific blendings which would, given the proper inputs, produce counterexamples to the Hirsch conjecture. We also show that the Hirsch conjecture is sharp for dimension 7.