Embedding convex geometries and a bound on convex dimension. / Richter, Michael; Rogers, Luke.

In: Discrete Mathematics, Vol. 340, No. 5, 05.2017, p. 1059–1063.

Research output: Contribution to journalArticlepeer-review

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Embedding convex geometries and a bound on convex dimension. / Richter, Michael; Rogers, Luke.

In: Discrete Mathematics, Vol. 340, No. 5, 05.2017, p. 1059–1063.

Research output: Contribution to journalArticlepeer-review

Harvard

Richter, M & Rogers, L 2017, 'Embedding convex geometries and a bound on convex dimension', Discrete Mathematics, vol. 340, no. 5, pp. 1059–1063. https://doi.org/10.1016/j.disc.2016.10.006

APA

Vancouver

Author

Richter, Michael ; Rogers, Luke. / Embedding convex geometries and a bound on convex dimension. In: Discrete Mathematics. 2017 ; Vol. 340, No. 5. pp. 1059–1063.

BibTeX

@article{1091fb5123564b479a299f547a62551b,
title = "Embedding convex geometries and a bound on convex dimension",
abstract = "The notion of an abstract convex geometry, due to Edelman and Jamison (1984), offers an abstraction of the standard notion of convexity in a linear space. Kashiwabara et al. (2005) 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 Edelman and Jamison (1984) and deduce a different upper bound on the dimension of the shelling. Furthermore, in the spirit of Cz{\'e}dli (2014), 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.",
author = "Michael Richter and Luke Rogers",
year = "2017",
month = may,
doi = "10.1016/j.disc.2016.10.006",
language = "English",
volume = "340",
pages = "1059–1063",
journal = "Discrete Mathematics",
issn = "0012-365X",
publisher = "Elsevier",
number = "5",

}

RIS

TY - JOUR

T1 - Embedding convex geometries and a bound on convex dimension

AU - Richter, Michael

AU - Rogers, Luke

PY - 2017/5

Y1 - 2017/5

N2 - The notion of an abstract convex geometry, due to Edelman and Jamison (1984), offers an abstraction of the standard notion of convexity in a linear space. Kashiwabara et al. (2005) 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 Edelman and Jamison (1984) and deduce a different upper bound on the dimension of the shelling. Furthermore, in the spirit of Czédli (2014), 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.

AB - The notion of an abstract convex geometry, due to Edelman and Jamison (1984), offers an abstraction of the standard notion of convexity in a linear space. Kashiwabara et al. (2005) 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 Edelman and Jamison (1984) and deduce a different upper bound on the dimension of the shelling. Furthermore, in the spirit of Czédli (2014), 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.

U2 - 10.1016/j.disc.2016.10.006

DO - 10.1016/j.disc.2016.10.006

M3 - Article

VL - 340

SP - 1059

EP - 1063

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 5

ER -