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J.-P. Roudneff conjectured in 1991 that every arrangement of n≥2d+1≥5 pseudohyperplanes in the real projective space Pd has at most ∑i=0d−2(n−1i) 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. Our main contribution is to show the validity of Roudneff's conjecture for d = 4. Moreover, based on computational data we show that for d≤4 and n≤2d+1 the maximum number of complete cells is only obtained (up to isomorphism) by cyclic arrangements.
Hernández-Ortiz et al. (Tue,) studied this question.