2017
DOI: 10.1016/j.disc.2016.10.006
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Embedding convex geometries and a bound on convex dimension

Abstract: 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 spir… Show more

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Cited by 16 publications
(33 citation statements)
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“…It was proved in [3] that every convex geometry of cdim = 2 is represented as (X, ch c ), for some set X of circles on the plane, but it was also noted that, in effect, just segments on a line are needed for representation. This observation was also reinforced in [7], where the representation by segments was realized in the spirit of Theorem 8. We give an example of representation for cdim = 2 used in [7] in the following example, and then summarize these results in Theorem 17 for completeness of the argument.…”
Section: Geometries Of Convex Dimensionmentioning
confidence: 70%
See 3 more Smart Citations
“…It was proved in [3] that every convex geometry of cdim = 2 is represented as (X, ch c ), for some set X of circles on the plane, but it was also noted that, in effect, just segments on a line are needed for representation. This observation was also reinforced in [7], where the representation by segments was realized in the spirit of Theorem 8. We give an example of representation for cdim = 2 used in [7] in the following example, and then summarize these results in Theorem 17 for completeness of the argument.…”
Section: Geometries Of Convex Dimensionmentioning
confidence: 70%
“…This observation was also reinforced in [7], where the representation by segments was realized in the spirit of Theorem 8. We give an example of representation for cdim = 2 used in [7] in the following example, and then summarize these results in Theorem 17 for completeness of the argument.…”
Section: Geometries Of Convex Dimensionmentioning
confidence: 70%
See 2 more Smart Citations
“…Interestingly enough, besides abstract convex geometry, the present work is motivated mainly by lattice theory. For more about the background and motivation of this topic, the reader may want, but need not, to see, for example, Adaricheva [1], Adaricheva and Czédli [3], Adaricheva and Nation [4] and [5], Czédli [7], [8], and [9], Czédli and Kincses [10], Edelman and Jamison [12], Kashiwabara, Nakamura, and Okamoto [14], Monjardet [17], and Richter and Rogers [19]. Note that the property described in 1.1(ii) is called the "Weak Carousel property" in Adaricheva and Bolat [2].…”
Section: Aim and Introductionmentioning
confidence: 99%