Roudneff's Conjecture for Lawrence Oriented Matroids

Abstract

J.-P. Roudneff has conjectured that every arrangement of $n\ge 2d+1\ge 5$ (pseudo) hyperplanes in the real projective space $\mathbb{P}^d$ has at most $\sum_{i=0}^{d-2} \binom{n-1}{i}$ cells bounded by each hyperplane. In this note, we show the validity of this conjecture for arrangements arising from Lawrence oriented matroids.