New results on $k$-Roudneff's conjecture

In this paper we study the number of $k$-neighborly reorientations of an oriented matroid, leading to study $k$-Roudneff's conjecture, the case $k=1$ being the original statement conjectured in 1991. We first prove the conjecture for the family of Lawrence oriented matroids (LOMs) with even rank $r=2k+2$ and also for low ranks by computer. Next, we provide a general upper bound for the number of $k$-neighborly reorientations of any LOM. Finally, we prove that for any $k\ge 1$ and any oriented matroid on $n$ elements, $k$-Roudneff's conjecture holds asymptotically as $n\rightarrow \infty$ and thus giving more credit to the conjecture.