The affine representation theorem for abstract convex geometries

Kashiwabara, Kenji; Nakamura, Masataka; Okamoto, Yoshio

doi:10.1016/j.comgeo.2004.05.001

Cited by 29 publications

(37 citation statements)

References 16 publications

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“…In this paper we provide an alternate proof of this representation theorem using a representation of [7] which represents a convex geometry through a collection of orderings. An important feature of this new proof is that it yields an upper bound on the dimension of the smallest Euclidean space into which a convex geometry may be embedded via a generalized convex shelling which may be different to the upper bound found by [13]. We then further use the representation theorem of [7] to provide a new result, that any convex geometry may be embedded as convex polygons in R 2 .…”

Section: Introductionmentioning

confidence: 94%

Embedding convex geometries and a bound on convex dimension

Richter

Rogers

2017

Discrete Mathematics

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The notion of an abstract convex geometry, due to Edelman and Jamison [7], offers an abstraction of the standard notion of convexity in a linear space. Kashiwabara, Nakamura, and Okamoto [13] introduce the notion of a generalized convex shelling into R N and prove that a convex geometry may always be represented with such a shelling. We provide a new, shorter proof of their result using a representation theorem of [7] and deduce a different upper bound on the dimension of the shelling. Furthermore, in the spirit of Czédli [5], who shows that any 2-dimensional convex geometry may be embedded as circles in R 2 , we show that any convex geometry may be embedded as convex polygons in R 2 .

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Section: Introductionmentioning

confidence: 94%

Embedding convex geometries and a bound on convex dimension

Richter

Rogers

2017

Discrete Mathematics

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“…Note that most of the finite convex geometries are not isomorphic to any of these restrictions. The first result that represents every finite convex geometry with the help of R n ; Conv R n was proved in Kashiwabara, Nakamura, and Okamoto [13]. This result uses auxiliary points and has not much to do with restrictions in our sense, so we do not give further details on it.…”

Section: Introductionmentioning

confidence: 99%

Representing convex geometries by almost-circles

Czédli¹,

Kincses²

2017

Acta Sci. Math.

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Finite convex geometries are combinatorial structures. It follows from a recent result of M. Richter and L.G. Rogers that there is an infinite set T RR of planar convex polygons such that T RR with respect to geometric convex hulls is a locally convex geometry and every finite convex geometry can be represented by restricting the structure of T RR to a finite subset in a natural way. An almost-circle of accuracy 1 − ǫ is a differentiable convex simple closed curve S in the plane having an inscribed circle of radius r 1 > 0 and a circumscribed circle of radius r 2 such that the ratio r 1 /r 2 is at least 1−ǫ. Motivated by Richter and Rogers' result, we construct a set Tnew such that (1) Tnew contains all points of the plane as degenerate singleton circles and all of its non-singleton members are differentiable convex simple closed planar curves;(2) Tnew with respect to the geometric convex hull operator is a locally convex geometry; (3) as opposed to T RR , Tnew is closed with respect to nondegenerate affine transformations; and (4) for every (small) positive ǫ ∈ R and for every finite convex geometry, there are continuum many pairwise affinedisjoint finite subsets E of Tnew such that each E consists of almost-circles of accuracy 1−ǫ and the convex geometry in question is represented by restricting the convex hull operator to E. The affine-disjointness of E 1 and E 2 means that, in addition to E 1 ∩ E 2 = ∅, even ψ(E 1 ) is disjoint from E 2 for every non-degenerate affine transformation ψ. Date: August 23, 2016. 1991 Mathematics Subject Classification. Primary 05B25; Secondary 06C10, 52A01. Key words and phrases. Abstract convex geometry, anti-exchange system, differentiable curve, almost-circle. This research was supported by NFSR of Hungary (OTKA), grant number K 115518. 1 (1.3) Points(X) = C∈X C, and Conv T (X) : = {D ∈ T : D ⊆ Conv R 2 (Points(X))}.

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“…Lower bounded lattices play an important role in lattice theory, especially in the study of free lattices, see monograph R. Freese, J. Ježek and J.B. Nation [12]. Finally, it was shown by K. Kashiwabara, M. Nakamura and Y. Okamoto [13] that every finite convex geometry is a sub-geometry of some affine convex geometry in R n , solving Problem 1.2 in the positive.…”

Section: Introductionmentioning

confidence: 99%

Representation of convex geometries by circles on the plane

Adaricheva¹,

Bolat²

2019

Discrete Mathematics

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Convex geometries are closure systems satisfying the anti-exchange axiom. Every finite convex geometry can be embedded into a convex geometry of finitely many points in an n-dimensional space equipped with a convex hull operator, by the result of K. Kashiwabara, M. Nakamura and Y. Okamoto (2005). Allowing circles rather than points, as was suggested by G. Czédli (2014), may presumably reduce the dimension for representation. This paper introduces a property, the Weak 2 × 3-Carousel rule, which is satisfied by all convex geometries of circles on a plane, and we show that it does not hold in all finite convex geometries. This raises a number of representation problems for convex geometries, which may allow us to better understand the properties of Euclidean space related to its dimension.

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The affine representation theorem for abstract convex geometries

Cited by 29 publications

References 16 publications

Embedding convex geometries and a bound on convex dimension

Embedding convex geometries and a bound on convex dimension

Representing convex geometries by almost-circles

Representation of convex geometries by circles on the plane

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