Roudneff's Conjecture in Dimension 4

03/24/2023

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by   Rangel Hernández-Ortiz, et al.

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J.-P. Roudneff conjectured in 1991 that every arrangement of n ≥ 2d+1≥ 5 pseudohyperplanes in the real projective space ℙ^d has at most ∑_i=0^d-2n-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.