Pushing the boundaries of polytopal realizability

Abstract

Abstract Let ∆(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 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 ∆(4, 10) = 5 and ∆(5, 11) = 6, and more recently, Bremner and Schewe showed ∆(4, 11) = ∆(6, 12) = 6. Here we show that ∆(4, 12) = ∆(5, 12) = 7 and present strong evidence that ∆(6, 13) = 7