  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 on complete
cells in arrangements. Adding to results of Larman and Garc\'ia-Col\'in, we
provide new bounds on the $k$-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 analyse. As a
consequence we prove the conjecture for odd rank $r$ and $k=\frac{r-1}{2}$ as
well as for rank $6$ and $k=2$ with the aid of the computer.
