Neighborly Families of Congruent Convex Polytopes
Joseph Zaks · American Mathematical Monthly · 1987
A family of convex d-polytopes in Ed is called neighborly [5], [6], [10]-[16] if every two members meet in a (d 1)-dimensional set (which lies, therefore, on a hyperplane; this hyperplane separates them and contains a facet of each one of them). A neighborly family in Ed, d > 3, can be infinite [3], [9]; however, if every member in a neighborly family has at most k facets, then there can be at most 2k members [8]. We [16], [15] have recently solved Bagemihl's conjecture ([1], see also [2], [5], [6]), which states that the maximum number of neighborly tetrahedra in E3 is 8; our proof depends heavily on Baston [2] and on a few searches by computer (Baston showed that the maximum is at most 9). The best known upper bound for neighborly families of d-simplices in Ed for d> 3 is 2dl1 [8]; the best known lower bound is 2d [11]. Concerning neighborly families of translates of a given convex polytope, there is an established maximum of 5 in E3 [4] and an upper bound of 2d 1 for all d > 4, which is conjectured to be also the maximum [4]. If all the members of a neighborly family are translates of a d-cube in Ed, then the maximum number of members is d + 1 [12]; but the following problem is open [7, #55]; [13].