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.
Highlights
An Euclidean d-arrangement of n hyperplanes H(d, n) is a finite collection of hyperplanes in the Euclidean space Rd such that no point belongs to every hyperplane of H(d, n)
We show the validity of this conjecture for arrangements arising from Lawrence oriented matroids
The cyclic polytope of dimension d with n vertices Cd(t1, . . . , tn), which was discovered by Caratheodory [2], is the convex hull in Rd, d 2 of n d + 1 different points x(t1), . . . , x(tn) of the moment curve x : R −→ Rd, t → (t, t2, . . . , td)
Summary
An Euclidean (resp. projective) d-arrangement of n hyperplanes H(d, n) is a finite collection of hyperplanes in the Euclidean space Rd (resp. the real projective space Pd) such that no point belongs to every hyperplane of H(d, n). In [10] Roudneff proved that the number of complete cells of the cyclic arrangements d−2 on dimension d with n hyperplanes, denoted as f (d, n), is at least n−1 i. Cyclic arrangements of n hyperplanes in Pd are equivalent to alternating oriented matroids of rank r = d + 1 on n elements. By using this combinatorial description, we may reformulate (1) as follows. Is it true that every arrangement with n r 3 hyperplanes in Pr−1 has at most f (r, n) complete cells ?. Any arrangement with n r 3 hyperplanes in Pr−1 arising from an acyclic Lawrence oriented matroid has at most f (r, n) complete cells. As we will see below, the class of Lawrence arrangements contains as a very particular case all the cyclic arrangements and a natural class to investigate the validity of the above question
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