2026-07-23T03:28:03Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:5102 2024-03-06T10:35:55Z ddc:004 open_access

Realization Spaces of Arrangements of Convex Bodies Dobbins, Michael Gene Holmsen, Andreas Hubard, Alfredo Oriented matroids Convex sets Realization spaces Mnev’s universality theorem 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Michael Gene Dobbins and Andreas Holmsen and Alfredo Hubard 2015 Is Part Of LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SOCG.2015.599 urn:nbn:de:0030-drops-51020 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.599 eng https://creativecommons.org/licenses/by/3.0/legalcode