More bounds on the diameters of convex polytopes

Let Δ(d, n) be the maximum possible diameter of the vertex-edge graph over all d-dimensional polytopes defined by n inequalities. The Hirsch bound holds for particular n and d if Δ(d, n)≤n-d. Francisco Santos recently resolved a question open for more than five decades by showing that Δ(d, 2d)≥d+1 for d=43; the dimension was then lowered to 20 by Matschke, Santos and Weibel. This progress has stimulated interest in related questions. The existence of a polynomial upper bound for Δ(d, n) is still an open question, the best bound being the quasi-polynomial one due to Kalai and Kleitman in 1992. Another natural question is for how large n and d the Hirsch bound holds. Goodey showed in 1972 that Δ(4, 10)=5 and Δ(5, 11)=6, and more recently, Bremner and Schewe showed that Δ(4, 11)=Δ(6, 12)=6. Here, we show that Δ(4, 12)=Δ(5, 12)=7.