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DOI: 10.4230/LIPIcs.SOCG.2015.599
URN: urn:nbn:de:0030-drops-51020
URL: http://drops.dagstuhl.de/opus/volltexte/2015/5102/
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Dobbins, Michael Gene ; Holmsen, Andreas ; Hubard, Alfredo

Realization Spaces of Arrangements of Convex Bodies

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Abstract

We introduce combinatorial types of arrangements of convex bodies, extending order types of point sets to arrangements of convex bodies, and study their realization spaces. Our main results witness a trade-off between the combinatorial complexity of the bodies and the topological complexity of their realization space. On one hand, we show that every combinatorial type can be realized by an arrangement of convex bodies and (under mild assumptions) its realization space is contractible. On the other hand, we prove a universality theorem that says that the restriction of the realization space to arrangements of convex polygons with a bounded number of vertices can have the homotopy type of any primary semialgebraic set.

BibTeX - Entry

@InProceedings{dobbins_et_al:LIPIcs:2015:5102,
  author =	{Michael Gene Dobbins and Andreas Holmsen and Alfredo Hubard},
  title =	{{Realization Spaces of Arrangements of Convex Bodies}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{599--614},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-83-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{34},
  editor =	{Lars Arge and J{\'a}nos Pach},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2015/5102},
  URN =		{urn:nbn:de:0030-drops-51020},
  doi =		{10.4230/LIPIcs.SOCG.2015.599},
  annote =	{Keywords: Oriented matroids, Convex sets, Realization spaces, Mnevís universality theorem}
}

Keywords: Oriented matroids, Convex sets, Realization spaces, Mnevís universality theorem
Seminar: 31st International Symposium on Computational Geometry (SoCG 2015)
Issue Date: 2015
Date of publication: 11.06.2015


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