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Realization Spaces of Arrangements of Convex Bodies

Authors Michael Gene Dobbins, Andreas Holmsen, Alfredo Hubard



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Michael Gene Dobbins
Andreas Holmsen
Alfredo Hubard

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Michael Gene Dobbins, Andreas Holmsen, and Alfredo Hubard. Realization Spaces of Arrangements of Convex Bodies. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 599-614, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)
https://doi.org/10.4230/LIPIcs.SOCG.2015.599

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.
Keywords
  • Oriented matroids
  • Convex sets
  • Realization spaces
  • Mnev’s universality theorem

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References

  1. Anders Björner, Michel Las Vergnas, Bernd Sturmfels, Neil White, and Günter M. Ziegler. Oriented matroids, volume 46 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1999. Google Scholar
  2. Raghavan Dhandapani, Jacob E. Goodman, Andreas Holmsen, and Richard Pollack. Interval sequences and the combinatorial encoding of planar families of pairwise disjoint convex sets. Rev. Roum. Math. Pures Appl, 50(5-6):537-553, 2005. Google Scholar
  3. Michael Gene Dobbins, Andreas Holmsen, and Alfredo Hubard. Regular systems of paths and families of convex sets in convex position. To appear in Transactions of the AMS. Google Scholar
  4. Michael Gene Dobbins, Andreas Holmsen, and Alfredo Hubard. The Erdős-Szekeres problem for non-crossing convex sets. Mathematika, 60(2):463-484, 2014. Google Scholar
  5. Stefan Felsner and Pavel Valtr. Coding and counting arrangements of pseudolines. Discrete & Computational Geometry, 46(3):405-416, 2011. Google Scholar
  6. Jon Folkman and Jim Lawrence. Oriented matroids. Journal of Combinatorial Theory, Series B, 25(2):199-236, 1978. Google Scholar
  7. Jacob E. Goodman. Proof of a conjecture of Burr, Grünbaum, and Sloane. Discrete Mathematics, 32(1):27-35, 1980. Google Scholar
  8. Jacob E. Goodman and Richard Pollack. On the combinatorial classification of nondegenerate configurations in the plane. Journal of Combinatorial Theory, Series A, 29(2):220-235, 1980. Google Scholar
  9. Jacob E. Goodman and Richard Pollack. Semispaces of configurations, cell complexes of arrangements. Journal of Combinatorial Theory, Series A, 37(3):257-293, 1984. Google Scholar
  10. Jacob E. Goodman and Richard Pollack. Upper bounds for configurations and polytopes in ℝ^d. Discrete & Computational Geometry, 1(1):219-227, 1986. Google Scholar
  11. Jacob E. Goodman and Richard Pollack. The combinatorial encoding of disjoint convex sets in the plane. Combinatorica, 28(1):69-81, 2008. Google Scholar
  12. Helmut Groemer. Geometric applications of Fourier series and spherical harmonics, volume 61 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1996. Google Scholar
  13. Branko Grünbaum. Arrangements and spreads. American Mathematical Society, 1972. Google Scholar
  14. Luc Habert and Michel Pocchiola. Arrangements of double pseudolines. In Proceedings of the 25th Annual Symposium on Computational Geometry, pages 314-323. ACM, 2009. Google Scholar
  15. Alfredo Hubard. Erdős-Szekeres para convexos. Bachelor’s thesis, UNAM, 2005. Google Scholar
  16. Alfredo Hubard, Luis Montejano, Emiliano Mora, and Andrew Suk. Order types of convex bodies. Order, 28(1):121-130, 2011. Google Scholar
  17. Michael Kapovich and John J. Millson. Universality theorems for configuration spaces of planar linkages. Topology, 41(6):1051-1107, 2002. Google Scholar
  18. Donald E. Knuth. Axioms and hulls, volume 606 of Lecture Notes in Computer Science. Springer-Verlag, 1992. Google Scholar
  19. Friedrich Levi. Die teilung der projektiven ebene durch gerade oder pseudogerade. Ber. Math.-Phys. Kl. Sächs. Akad. Wiss, 78:256-267, 1926. Google Scholar
  20. Nicolai E. Mnev. Varieties of combinatorial types of projective configurations and convex polytopes. Doklady Akademii Nauk SSSR, 283(6):1312-1314, 1985. Google Scholar
  21. Nicolai E. Mnev. The universality theorems on the classification problem of configuration varieties and convex polytopes varieties. In Topology and Geometry: Rohlin seminar, pages 527-543. Springer, 1988. Google Scholar
  22. Mordechai Novick. Allowable interval sequences and line transversals in the plane. Discrete & Computational Geometry, 48(4):1058-1073, 2012. Google Scholar
  23. Mordechai Novick. Allowable interval sequences and separating convex sets in the plane. Discrete & Computational Geometry, 47(2):378-392, 2012. Google Scholar
  24. János Pach and Géza Tóth. Families of convex sets not representable by points. In Algorithms, architectures and information systems security, volume 3, page 43. World Scientific, 2008. Google Scholar
  25. Arnau Padrol and Louis Theran. Delaunay triangulations with disconnected realization spaces. In Proceedings of the 30th Annual Symposium on Computational Geometry, pages 163-170. ACM, 2014. Google Scholar
  26. Jürgen Richter-Gebert. Realization spaces of polytopes, volume 1643 of Lecture Notes in Mathematics. Springer Verlag, 1996. Google Scholar
  27. Gerhard Ringel. Teilungen der ebene durch geraden oder topologische geraden. Mathematische Zeitschrift, 64(1):79-102, 1956. Google Scholar
  28. Peter W. Shor. Stretchability of pseudolines is NP-hard. In Applied Geometry and Discrete Mathematics: The Victor Klee Festschrift, volume 4, pages 531-554. American Mathematical Society, 1991. Google Scholar
  29. Yasuyuki Tsukamoto. New examples of oriented matroids with disconnected realization spaces. Discrete & Computational Geometry, 49(2):287-295, 2013. Google Scholar
  30. Ravi Vakil. Murphy’s law in algebraic geometry: badly-behaved deformation spaces. Inventiones Mathematicae, 164(3):569-590, 2006. Google Scholar
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