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.
翻译：J.-P. Roudneff 在 1991 年提出了猜想：在实射影空间 $\mathbb{P}^d$ 中，$n\ge 2d+1\ge 5$ 个次超平面的任意排列最多包含 $\sum_{i=0}^{d-2} \binom{n-1}{i}$ 个完全胞腔（即由每个超平面所限定的胞腔）。对于 $d=2,3$ 和 Lawrence 定向拟阵的排列，该猜想是正确的。本文的主要结果是证明了当 $d=4$ 时，Roudneff 猜想的正确性。此外，基于计算数据，我们猜想完全胞腔的最大数量仅在循环排列中获得。
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