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arXiv · 2303.14212

Roudneff's Conjecture in Dimension $4$

Rangel Hernández-Ortiz·Kolja Knauer·Luis Pedro Montejano·Manfred Scheucher
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Abstract

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.

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Rangel Hernández-Ortiz, Kolja Knauer, Luis Pedro Montejano, Manfred Scheucher. 2023-03-24. Roudneff's Conjecture in Dimension $4$. https://arxiv.org/abs/2303.14212

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