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

Roudneff, Jean-Pierre

doi:10.1016/0012-365x(86)90016-6

Cited by 28 publications

(25 citation statements)

References 5 publications

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“…The main goal of the present paper is to prove Gru nbaum's conjecture: We point out that Theorem 1.2 is in a certain sense best possible. Indeed, we have proved in [10] that the equality p 3 = 1 3 n(n&1) is achieved for infinitely many simple arrangements (an alternative construction of such arrangements has also been obtained by Harborth [7]). Moreover, the value n 0 =9 in Theorem 1.2 cannot be decreased.…”

Section: A General Upper Bound For the Number Of Trianglesmentioning

confidence: 95%

“…Let R(4k+2) denote the regular simplicial arrangement of n=4k+2 lines, consisting of the 2k+1 lines determined by the edges of a regular (2k+1)-gon, together with the 2k+1 axes of symmetry of that polygon ( [4, p. 9]). Assume that there exists a simple extremal arrangement A of 2k+2 pseudolines (this is actually possible for infinitely many values of k, see [7], [10]), and let L denote a pseudoline of A. We transform R(4k+2), using perturbation in the neighborhood of the center of the (2k+1)-gon, in such a way that the 2k+1 modified pseudolines form an arrangement isomorphic to A "L .…”

Section: Extremal Examplesmentioning

confidence: 99%

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The Maximum Number of Triangles in Arrangements of Pseudolines

Roudneff

1996

Journal of Combinatorial Theory, Series B

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Gru nbaum has conjectured that any arrangement of n pseudolines in the real projective plane has at most 1 3 n(n&1) triangular faces when n is sufficiently large. We prove this conjecture for n 9; the result does not hold for n 8. The structure of extremal examples is explored and an infinite family of non simple arrangements with 1 3 n(n&1) triangles is constructed. As an application, we show that the number of simplices in arrangements of n 10 pseudoplanes is always less than 1 12 n(n&1)(n&2).

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Section: A General Upper Bound For the Number Of Trianglesmentioning

confidence: 95%

Section: Extremal Examplesmentioning

confidence: 99%

The Maximum Number of Triangles in Arrangements of Pseudolines

Roudneff

1996

Journal of Combinatorial Theory, Series B

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“…This situation changes rapidly if we focus our attention on convexity properties of point configurations. The theory of oriented matroids or chirotopes -see, e.g., [4], [6], [8], [21], [28] -has shown that point configurations in general position also have a rather rich and complicated structure, a fact that is well-known in the geometry of convexity. Let us mention in this context a result of Bokowski and Sturmfels: there is no local criterion to decide whether a uniform rank 3 oriented matroid is representable (eoordinatizable) [7].…”

Section: Mutations Of Simplicial Chirotopesmentioning

confidence: 99%

Simplicial cells in arrangements and mutations of oriented matroids

Roudneff

Sturmfels

1988

Geom Dedicata

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We investigate the combinatorial and topological properties of simplicial cells in arrangements of (pseudo)hyperplanes, using their interpretations in terms of oriented matroids. Simplicial cells have various applications in computational geometry due to the fact that for an arrangement in general position they are in one-to-one correspondence to local changes ('mutations') of its combinatorial type. Several characterizations for mutations of oriented matroids, and their relation to geometric realizability questions are being discussed.We give two short proofs for a result of Shannon [30] that every arrangement of n hyperplanes in E d has at least n simplicial cells, this bound being sharp for every n and d. The concatenation operation, a construction introduced by Lawrence and Weinberg [21], will be used to generate a large class of representable oriented matroids with this minimal number of mutations.A homotopy theorem is proved, stating that any two arrangements in general position can be transformed into each other be a sequence of representability preserving mutations. Finally, we give an example of an oriented matroid on eight elements with only seven mutations. As a corollary we obtain a new proof for the non-polytopality of the smallest non-polytopal matroid sphere M9963.

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“…This proves the following well-known theorem. Theorem 1.1 [3], [4]. Let .4 be a simple p3-maximal arrangement of n > 4 pseudolines.…”

Section: Introductionmentioning

confidence: 99%

“…There are known recursive methods by Roudneff [4] and Harborth [3] to construct a simple p3-maximal arrangement with 2(n -1) pseudolines from a simple p3-maximal arrangement with n pseudolines.…”

Section: Introductionmentioning

confidence: 99%