Construction and Analysis of Projected Deformed Products

We introduce a deformed product construction for simple polytopes in terms of lowertriangular block matrix representations. We further show how Gale duality can be employed for the construction and for the analysis of deformed products such that specified faces (e.g. all the k-faces) are “strictly preserved” under projection. Thus, starting from an arbitrary neighborly simplicial (d−2)-polytope Q on n−1 vertices we construct a deformed n-cube, whose projection to the last d coordinates yields a neighborly cubical d-polytope. As an extension of the cubical case, we construct matrix representations of deformed products of (even) polygons (DPPs), which have a projection to d-space that retains the complete (⌊ 2 ⌋ − 1)-skeleton. In both cases the combinatorial structure of the images under projection is completely determined by the neighborly polytope Q: Our analysis provides explicit combinatorial descriptions. This yields a multitude of combinatorially different neighborly cubical polytopes and DPPs. As a special case, we obtain simplified descriptions of the neighborly cubical polytopes of Joswig & Ziegler (2000) as well as of the projected deformed products of polygons that were announced by Ziegler (2004), a family of 4-polytopes whose “fatness” gets arbitrarily close to 9.