Roudneff's Conjecture in Dimension $4$

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 ${\sum }_{i=0}^{d-2}\left(\genfrac{}{}{0px}{}{n-1}{i}\right)$ 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.