as the diameter of a polytope provides a lower bound on the number of steps needed by the simplex method. The conjecture is now known to be false in general.

The Hirsch conjecture was proven for d < 4 and for various special cases, while the best known upper bounds on the diameter are only sub-exponential in n and d. After more than fifty years, a counter-example was announced in May 2010 by Francisco Santos Leal, from the University of Cantabria. The result was presented at the conference 100 Years in Seattle: the mathematics of Klee and Grünbaum and appeared in Annals of Mathematics. Specifically, the paper presented a 43-dimensional polytope of 86 facets with a diameter of more than 43. The counterexample has no direct consequences for the analysis of the simplex method, as it does not rule out the possibility of a larger but still linear or polynomial number of steps.