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Realization Spaces of Arrangements of Convex Bodies

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.
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https://hal.inria.fr/hal-01203785
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Submitted on : Wednesday, September 23, 2015 - 5:39:51 PM
Last modification on : Wednesday, February 26, 2020 - 7:06:07 PM
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Michael Gene Dobbins, Andreas Holmsen, Alfredo Hubard. Realization Spaces of Arrangements of Convex Bodies. Symposium on Computational Geometry 2015, Jun 2015, Eindhoven, Netherlands. pp.16, ⟨10.4230/LIPIcs.SOCG.2015.599⟩. ⟨hal-01203785⟩

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