Roudneff’s Conjecture in Dimension 4

Abstract

J.-P. Roudneff conjectured in 1991 that every arrangement of n ≥ 2 d + 1 ≥ 5 pseudohyperplanes in the real projective space P d has at most ∑ i = 0 d − 2 ( 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. 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 ≤ 2 d + 1 the maximum number of complete cells is only obtained (up to isomorphism) by cyclic arrangements.