Combinatorial Abstractions for the Diameter of Polytopes Paderborn, Den 2. Oktober 2008

Polytopes Abstract Polyhedra Ultraconnected Families Abstract Polyhedral Graphs Blueprints It is unclear which of the inequalities shown in this diagram are strict. What we do know for a fact is that for dimension d ≤ 5, the maximum diameter of APGs is strictly larger than the maximum diameter of abstract polytopes (this follows from [AD74] and theorem 5.4.2), and for dimension 4 and 5 the maximum diameter of polyhedra is strictly larger than the maximum diameter of polytopes. Lemma 3.4.2. Let V be an ultraconnected family of d-sets. Then there is a partition L = (L0, . . . ,L`) of V such that L is an abstract polyhedral graph and diam(V ) = diam(L ) = `. Proof. Let v, w ∈ V be two vertices such that d(v, w) = diam(V ). Partition the vertices of V according to their distance from v , that is, L j := {u ∈ V | d(u, v) = j } for all 0 ≤ j ≤ ` = di am(V ). Every vertex in V has a unique distance to v less than or equal to diam(V ), so L = (L0, . . . ,L`) partitions V .