  J.-P. Roudneff conjectured in 1991 that every arrangement of $n \ge 2d+1\ge
5$ pseudohyperplanes in the real projective space $\mathbb{P}^d$ has at most
$\sum_{i=0}^{d-2} \binom{n-1}{i}$ complete cells (i.e., cells bounded by each
hyperplane). The conjecture is true for $d=2,3$ and for arrangements arising
from Lawrence oriented matroids. The main result of this manuscript is to show
the validity of Roudneff's conjecture for $d=4$. Moreover, based on
computational data we conjecture that the maximum number of complete cells is
only obtained by cyclic arrangements.
