Construction and Analysis of Projected Deformed Products

We introduce a deformed product construction for simple polytopes in terms of lower-triangular block matrix representations. We further show how Gale duality can be employed for the construction and 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 -skeleton. In both cases the combinatorial structure of the images under projection is determined by the neighborly simplicial 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 and Ziegler (Discrete Comput. Geom. 24:325–344, 2000 ) as well as of the projected deformed products of polygons announced by Ziegler (Electron. Res. Announc. Am. Math. Soc. 10:122–134, 2004 ), a family of 4-polytopes whose “fatness” gets arbitrarily close to 9.