Bounds on the cardinalities of nearly neighborly and neighborly families of polytopes
A family of polytopes in E$\sp{d}$ is called nearly neighborly if, for every two members of the family, there is a hyperplane which separates them and contains a facet of each. Such a family is called neighborly if every two members of the family have a $(d - 1)$-dimensional intersection. We have found nearly neighborly families of nine quadrilaterals and conjecture that this is the maximum. Using techniques developed by J. Zaks for nearly-neighborly tetrahedra, we show that a family of nearly neighborly quadrilaterals has at most 14 members. A family of nearly neighborly quadrilaterals is said to share a base line if all nearly neighborly quadrilaterals lie on the same side of the line and the line contains a side of each. These families form an important special case. We show that there are 35 inequivalent examples of four nearly neighborly quadrilaterals on a base line, but that it is impossible to have five nearly neighborly quadrilaterals on a base line. We also consider related questions about other two- and three-dimensional figures. We prove that the maximum cardinality of a nearly neighborly family of triangles is four. We present maximal nearly neighborly families of unit squares and rectangles. We show examples of nearly neighborly families of unit cubes and rectangular parallelepipeds. We also give some results about nearly neighborly families of nonconvex polytopes. We continue with some results for neighborly families of polytopes including a proof that there can be at most four polygons (with any number of sides) in a neighborly family in E$\sp2$. We conclude with some questions for further study.
University of Illinois - Urbana-Champaign
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University of Illinois at Urbana-Champaign