投影变形产品的构造与分析
Construction and Analysis of Projected Deformed Products
摘要：
本文利用下三角块矩阵表示，给出了简单多面体的变形积构造。我们进一步展示了如何盖尔对偶可以采用的建设和变形产品的分析，使指定的面（例如所有的k-面）是“严格的”下投影。因此，从n-1个顶点上的任意邻单（d-2）-多面体Q出发，我们构造了一个变形的n-立方体，其投影到最后一个d坐标产生一个邻三次d-多面体。作为立方情形的推广，我们构造了（偶）多边形的变形积（DPP）的矩阵表示，这些变形积在d-空间的投影保持了完整的（\lfloor \tfrac{d}{2} \rfloor - 1）-骨架.在这两种情况下的组合结构的图像下的投影是完全由相邻的多面体Q：我们的分析提供了明确的组合描述。这产生了大量的组合不同的相邻立方多面体和DPP。作为一个特例，我们得到Joswig & Ziegler（2000）的相邻立方多面体的简化描述，以及Ziegler（2004）宣布的"投影变形的乘积“，一个4-多面体族，其"胖度”任意接近9。
英文摘要：
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 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 dcoordinates yields a neighborly cubical d-polytope. As an extension of thecubical 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 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.
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