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Realization Spaces of Arrangements of Convex Bodies

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Abstract

We introduce combinatorial types of planar 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. First, we show that every combinatorial type is realizable and its realization space is contractible under mild assumptions. Second, we prove a universality theorem that says the restriction of the realization space to arrangements polygons with a bounded number of vertices can have the homotopy type of any primary semialgebraic set.

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Notes

  1. Note that Mnev’s Theorem is more specific as it deals with stable equivalence.

  2. The definition of \(f_{ i}\) on \(\Theta (v)\) is irrelevant as long as \(f_{ i}\) is \(C^2\)-smooth, monotonic, symmetric about \(\theta \), and varies continuously with respect to V. A cubic spline would suffice for this.

  3. Here subscripts are indices over \(\mathbb {Z}_k\), so in particular \(\ell _k\) is the line spanning \(a_k^n\) and \(a_1^1\).

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Acknowledgements

M. G. Dobbins was supported by National Research Foundation Grant NRF-2011-0030044 (SRC-GAIA) funded by the government of South Korea. A. Holmsen was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2010-0021048). A. Hubard was supported by Fondation Sciences Mathématiques de Paris and by the Advanced Grant of the European Research Council GUDHI (Geometric Understanding in Higher Dimensions).

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Authors and Affiliations

  1. Department of Mathematical Sciences, Binghamton University, Binghamton, NY, 13902-6000, USA

    Michael Gene Dobbins

  2. Department of Mathematical Sciences, KAIST, 291 Daehak-ro Yuseong-gu, Taejon, 34141, South Korea

    Andreas Holmsen

  3. Université Paris-Est Marne-la-Vallée, 77454, Marne-la-Vallée Cedex 2, France

    Alfredo Hubard

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  1. Michael Gene Dobbins
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  2. Andreas Holmsen
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  3. Alfredo Hubard
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Correspondence to Michael Gene Dobbins.

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Dobbins, M.G., Holmsen, A. & Hubard, A. Realization Spaces of Arrangements of Convex Bodies. Discrete Comput Geom 58, 1–29 (2017). https://doi.org/10.1007/s00454-017-9869-8

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