Roudneff’s Conjecture in Dimension 4

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 ${\sum }_{i=0}^{d-2}(\begin{array}{c}n-1\\ i\end{array})$ 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 $d\le 4$ and $n\le 2d+1$ the maximum number of complete cells is only obtained (up to isomorphism) by cyclic arrangements.

Keywords:

• Roudneff’s conjecture
• oriented matroid, arrangement of hyperplanes

Mathematics Subject Classification:

• 52C40
• 05C35
• 05Dxx
• 68Rxx

Additional information

Funding

K. K. was supported by the Spanish State Research Agency through grants RYC-2017-22701, PID2019-104844GB-I00, PID2022-137283NB-C22 and the Severo Ochoa and María de Maeztu Program for Centers and Units of Excellence in R&D (CEX2020-001084-M). R. H. and L. P. M. were supported by the grant 2021 SGR 00115 from the Government of Catalonia, by the project ACITECH PID2021-124928NB-I00, funded by MCIN/AEI/10.13039/501100011033/FEDER, EU, and by the project HERMES, funded by INCIBE and by the European Union NextGeneration EU/PRTR. M. S. was supported by DFG Grant SCHE 2214/1-1.