Arrangements Of Pseudolines Research Articles

Reconstruction of The Orientation Class of An Oriented Matroid

We show that the orientation class of an oriented matroid of corank ⩾ 3 is completely determined by the list of orientation classes of its one-element deletions. This result contains as a special case (corank 3) a theorem of J. E. Goodman and R. Pollack on arrangements of pseudolines. As another application we give an affirmative answer to a problem of R. Cordovil and P. Duchet on a characterization of orientation classes. Extending a result of R. G. Bland and M. Las Vergnas, we also characterize regular (unimodular) matroids in terms of unique orientability.

Arrangements of lines with a minimum number of triangles are simple

Levi has shown that for every arrangement ofn lines in the real projective plane, there exist at leastn triangular faces, and Grunbaum has conjectured that equality can occur only for simple arrangements. In this note we prove this conjecture. The result does not hold for arrangements of pseudolines.

On the number of triangles in simple arrangements of pseudolines in the real projective plane

For any simple arrangement A of n⩾4 pseudolines in the realprojective plane, it is easy to show that the number P 3 of triangles in the 2-dimensional cell complex associated with A satisfies P 3⩽ 1 3 n( n−1). In this paper we construct infinite classes of simple arrangements for which P 3= 1 3 n( n-1), answering negatively a conjecture of Grünbaum. An interpretation of our results in terms of oriented matroids is also given.

Polynomial realization of pseudoline arrangements

AbstractAn arrangement of pseudolines is called d‐stretchable if it is isomorphic to an arrangement of graphs of polynomial functions, each of degree at most d; in particular, it is 1‐stretchable (classically: “stretchable”) if and only if it is isomorphic to an arrangement of lines. We show that (i) if every arrangement of n pseudolines is d‐stretchable, then d ≦ cn1/2, and (ii) every arrangement of n pseudolines is d‐stretchable with d = cn2.

SOME SIMPLE ARRANGEMENTS OF PSEUDOLINES WITH A MAXIMUM NUMBER OF TRIANGLES

SOME SIMPLE ARRANGEMENTS OF PSEUDOLINES WITH A MAXIMUM NUMBER OF TRIANGLES

Helly-type theorems for pseudoline arrangements in P 2

We prove duals of Radon's theorem, Helly's theorem, Carathéodory's theorem, and Kirchberger's theorem for arrangements of pseudolines in the real projective plane, which generalize the original versions of those theorems for plane configurations of points. We also prove a topological generalization of the pseudoline-dual of Helly's theorem.

Sur un theoreme de separation des matroïdes orientes de rang trois

We prove a separation theorem for oriented matroids of rank 3. We give short combinatorial proofs of Carathéodory's theorem for arrangements of pseudohyperplanes and Kirchberger's theorem for arrangements of pseudolines.

Proof of Grünbaum's conjecture on the stretchability of certain arrangements of pseudolines

We prove Grünbaum's conjecture that every arrangement of eight pseudolines in the projective plane is stretchable, i.e., determines a cell complex isomorphic to one determined by an arrangement of lines. The proof uses our previous results on ordered duality in the projective plane and on periodic sequences of permutations of [1, n] associated to arrangements of n lines in the euclidean plane.

Sylvester's Problem for Spreads of Curves

Spreads of curves were introduced by Grunbaum in [1]. A spread of curves is a continuous family of simple arcs in the real plane, every two of which intersect in exactly one point. A spread is the continuous analogue of a finite arrangement of pseudolines in the plane. Sylvester's problem for finite arrangements of pseudolines asks if every non-trivial arrangement has a simple vertex, that is a point contained in exactly two pseudolines of the arrangement. This question was answered in the affirmative by Kelly and Rottenberg [5]. One interesting feature of this result is that it does not depend on the pseudolines being straight lines.Here we settle Sylvester's problem for spreads. We show that every nontrivial spread of line segments has uncountably many simple vertices. But we also give examples of non-trivial spreads with no simple vertices.

Proof of a conjecture of Burr, Grünbaum, and Sloane

We establish a duality principle for arrangements of pseudolines in the projective plane, and thereby prove the conjecture of Burr, Grünbaum, and Sloane that the solution ▪ of the “orchard problem” for pseudoline arrangements and the solution t̃( p) of the dual problem are equal.

Simple points in pseudoline arrangements

Simple points in pseudoline arrangements