# Construction and Analysis of Projected Deformed Products

Discrete & Computational Geometry, Mar 2009

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 $(\lfloor\tfrac{d}{2}\rfloor-1)$ -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.

This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2Fs00454-009-9146-6.pdf

Raman Sanyal, Günter M. Ziegler. Construction and Analysis of Projected Deformed Products, Discrete & Computational Geometry, 2009, 412-435, DOI: 10.1007/s00454-009-9146-6