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Roudneff's Conjecture in Dimension $4$
Rangel Hernández-Ortiz, Kolja Knauer, Luis Pedro Montejano, Manfred Scheucher
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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. 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.
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