Finite Point Sets and Oriented Matroids Combinatorics in Geometry

Bokowski, Jürgen

doi:10.1007/978-1-4612-4088-4_4

Cited by 3 publications

(3 citation statements)

References 13 publications

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“…It follows that the facial structure of M * is still given by one of the drawings of Figure 2. Note that, by the results of Bokowski and Richter (see [6]), non-realizability actually occurs in exactly 24 cases.…”

Section: K-lopsided Oriented Matroidsmentioning

confidence: 91%

Partitions of Points into Simplices withk-dimensional Intersection. Part II: Proof of Reay’s Conjecture in Dimensions 4 and 5

Roudneff¹

2001

European Journal of Combinatorics

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A longstanding conjecture of Reay asserts that every set X of (m−1)(d +1)+k+1 points in general position in R d has a partition X 1 , X 2 , . . . , X m such that m i=1 conv X i is at least k-dimensional. Using the tools developed in [13] and oriented matroid theory, we prove this conjecture for d = 4 and d = 5. How about, to that end, we introduce the notion of a k-lopsided oriented matroid and we characterize these combinatorial objects for certain values of k. Divisibility properties for subsets of R d with other independence conditions are also obtained, thus settling several particular cases of a generalization of Reay's conjecture.

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Section: K-lopsided Oriented Matroidsmentioning

confidence: 91%

Partitions of Points into Simplices withk-dimensional Intersection. Part II: Proof of Reay’s Conjecture in Dimensions 4 and 5

Roudneff¹

2001

European Journal of Combinatorics

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“…A feasible combination of reaction directions naturally corresponds to a sign vector (having –, 0, or + entries) of the flux cone, and every FT corresponds to a support-maximal sign vector of the flux cone. In fact, the term ‘tope’ comes from the theory of oriented matroids, where it refers to a maximal sign vector of a linear subspace (Bachem and Kern, 1992; Bokowski, 2006). Whereas an EFM represents a minimal pathway (involving a minimal set of reactions), a FT contains a maximal ‘pathway’ (involving a maximal set of reactions).…”

Section: Introductionmentioning

confidence: 99%

Flux tope analysis: studying the coordination of reaction directions in metabolic networks

et al. 2018

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MotivationElementary flux mode (EFM) analysis allows an unbiased description of metabolic networks in terms of minimal pathways (involving a minimal set of reactions). To date, the enumeration of EFMs is impracticable in genome-scale metabolic models. In a complementary approach, we introduce the concept of a flux tope (FT), involving a maximal set of reactions (with fixed directions), which allows one to study the coordination of reaction directions in metabolic networks and opens a new way for EFM enumeration.ResultsA FT is a (nontrivial) subset of the flux cone specified by fixing the directions of all reversible reactions. In a consistent metabolic network (without unused reactions), every FT contains a ‘maximal pathway’, carrying flux in all reactions. This decomposition of the flux cone into FTs allows the enumeration of EFMs (of individual FTs) without increasing the problem dimension by reaction splitting. To develop a mathematical framework for FT analysis, we build on the concepts of sign vectors and hyperplane arrangements. Thereby, we observe that FT analysis can be applied also to flux optimization problems involving additional (inhomogeneous) linear constraints. For the enumeration of FTs, we adapt the reverse search algorithm and provide an efficient implementation. We demonstrate that (biomass-optimal) FTs can be enumerated in genome-scale metabolic models of B.cuenoti and E.coli, and we use FTs to enumerate EFMs in models of M.genitalium and B.cuenoti.Availability and implementationThe source code is freely available at https://github.com/mpgerstl/FTA.Supplementary information Supplementary data are available at Bioinformatics online.

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“…We start with the list of 135 reorientation classes of uniform rank 5 oriented matroids on eight elements [2,3]. From this list we can generate the 3501 non-isomorphic matroid polytopes of rank 5 with eight vertices.…”

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confidence: 99%

10 Points in Dimension 4 not Projectively Equivalent to the Vertices of a Convex Polytope

Forge¹,

Vergnas²,

Schuchert

2001

European Journal of Combinatorics

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Using oriented matroids, and with the help of a computer, we have found a set of 10 points in R 4 not projectively equivalent to the vertices of a convex polytope. This result confirms a conjecture of Larman [6] in dimension 4. c 2001 Academic Press PROBLEM (McMullen [6]). Determine the largest integer n = f (d) such that for any given n points in general position in R d there is an admissible projective transformation mapping these points onto the vertices of a convex polytope.Here admissible means that none of the n points is sent to infinity by the projective transformation.For dimension two and three the numbers f (d) are known: f (2) = 5 and f (3) = 7. For d ≥ 2, Larman has established in [6] the bounds 2dby Las Vergnas [7], as a corollary of Redei's theorem for tournaments. Recently, Ramírez Alfonsín [8] has proven the linear upper bound f (d) ≤ 5d/2 + 1, by a construction using Lawrence oriented matroids (unions of rank 1 oriented matroids).In the context of oriented matroids the problem can be conveniently restated in terms of hyperplanes. We refer the reader to [1] for information regarding oriented matroid theory. As easily seen, the oriented matroids of the images of a given configuration of points by admissible projective tranformations are all the acyclic reorientations of the oriented matroid defined by the affine dependencies of the configuration. The dual of a configuration of points is an arrangement of hyperplanes, and the regions defined by this arrangement are in 1-1 correspondence with the acyclic reorientations of the oriented matroid. We say that a region which meets all hyperplanes in dimension d − 1 is complete. It is almost immediate to verify that a region is complete if and only if all corresponding admissible projective transformations maps the given n points onto the set of vertices of convex polytopes (note that these convex polytopes necessarily have the same oriented matroid).Hence the McMullen problem is equivalent to: determine the largest integer n = f (d) such that any arrangement of n hyperplanes in general position in R d contains a complete region.The same problem for general oriented matroids has been considered by Cordovil and Da Silva [4]: determine the largest integer n = g(r ) such that any uniform rank r oriented matroid M with n elements has a complete region. A region (or tope) of an oriented matroid is a region determined by the pseudohyperplanes of its topological representation. The regions of an oriented matroid are in 1-1 correspondence with its maximal covectors, and a region is complete if and only if changing the sign of any element in the corresponding maximal covector produces another maximal covector. Obviously g(r ) ≤ f (r + 1). Cordovil and Da Silva have shown in [4] that 2r − 1 ≤ g(r ), generalizing Larman's lower bound.In this paper, we construct uniform rank 5 oriented matroids on 10 elements without complete region, hence, g(5) = 9. One of these oriented matroids has a realization in R 4 , hence f (4) = 9. † C.N.R.S., Paris.

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Finite Point Sets and Oriented Matroids Combinatorics in Geometry

Cited by 3 publications

References 13 publications

Partitions of Points into Simplices withk-dimensional Intersection. Part II: Proof of Reay’s Conjecture in Dimensions 4 and 5

Partitions of Points into Simplices withk-dimensional Intersection. Part II: Proof of Reay’s Conjecture in Dimensions 4 and 5

Flux tope analysis: studying the coordination of reaction directions in metabolic networks

10 Points in Dimension 4 not Projectively Equivalent to the Vertices of a Convex Polytope

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