Diameter of Polyhedra : Limits of Abstraction Friedrich Eisenbrand

We investigate the diameter of a natural abstraction of the 1-skeleton of polyhedra. Even if this abstraction is more general than other abstractions previously studied in the literature, known upper bounds on the diameter of polyhedra continue to hold here. On the other hand, we show that this abstraction has its limits by providing an almost quadratic lower bound. 1. Introduction. One of the most prominent mysteries in convex geometry is the question whether the diameter of polyhedra is polynomial in the number of its facets or not. If the largest diameter of a d-dimensional and possibly unbounded polyhedron with n facets is denoted by △ u (d, n), then the best known upper bound is △ u (d, n) ≤ n 1+log d , shown by Kalai and Kleitman [13]. For a long time, the best known lower bound was △ u (d, n) ≥ n − d + ⌊d/5⌋, due to Klee and Walkup [15]. Recently, Santos [20] has given a lower bound of △ u (d, n) ≥ (1 + ε)(n − d), where d, ǫ are fixed and n is arbitrarily large. The gap which is left open here is huge, even after decades of intensive research on this problem.