On k-neighborly reorientations of oriented matroids

We study the existence and the number of k-neighborly reorientations of an oriented matroid. This leads to k-variants of McMullen’s problem and Roudneff’s conjecture, the case k=1 being the original statements. Adding to results of Larman and García-Colín, we provide new bounds on k-McMullen’s problem and prove the conjecture for several ranks and k by computer. Further, we show that k-Roudneff’s conjecture for fixed rank and k reduces to a finite case analysis. As a consequence we prove the conjecture for odd rank r and k=r−12 as well as for rank 6 and k=2 with the aid of the computer.