The Monotonic Bounded Hirsch Conjecture is False for Dimension at Least 4

We exhibit a d -polytope P with n facets and a linear form (phi) such that all paths from a certain vertex of P to a (phi)-minimizing vertex that are nonincreasing in (phi) have length at least n - d + 1, provided d (ge) 4 and n - d (ge) 4. Indeed the discrepancy between the minimum length of such paths and n - d is at least min{[ d /4], [( n - d )/4]} for some such polytope.