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Universality Theorems for Inscribed Polytopes and Delaunay Triangulations

Abstract

We prove that every primary basic semi-algebraic set is homotopy equivalent to the set of inscribed realizations (up to Möbius transformation) of a polytope. If the semi-algebraic set is, moreover, open, it is, additionally, (up to homotopy) the retract of the realization space of some inscribed neighborly (and simplicial) polytope. We also show that all algebraic extensions of \({\mathbb {Q}}\) are needed to coordinatize inscribed polytopes. These statements show that inscribed polytopes exhibit the Mnëv universality phenomenon. Via stereographic projections, these theorems have a direct translation to universality theorems for Delaunay subdivisions. In particular, the realizability problem for Delaunay triangulations is polynomially equivalent to the existential theory of the reals.

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References

  1. Adiprasito, K.A., Padrol, A.: The universality theorem for neighborly polytopes. Combinatorica (to appear). arXiv:1402.7207

  2. Adiprasito, K.A., Ziegler, G.M.: Many projectively unique polytopes. Invent. Math. 199(3), 581–652 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  3. Björner, A., Las Vergnas, M., Sturmfels, B., White, N., Ziegler, G.M.: Oriented Matroids. Encyclopedia of Mathematics and Its Applications, vol. 46, 2nd edn. Cambridge University Press, Cambridge (1999)

    Google Scholar 

  4. Bokowski, J., Guedes de Oliveira, A.: Simplicial convex 4-polytopes do not have the isotopy property. Port. Math. 47(3), 309–318 (1990)

    MathSciNet  MATH  Google Scholar 

  5. Bokowski, J., Ewald, G., Kleinschmidt, P.: On combinatorial and affine automorphisms of polytopes. Isr. J. Math. 47(2–3), 123–130 (1984)

    MathSciNet  Article  MATH  Google Scholar 

  6. Brown, K.Q.: Voronoi diagrams from convex hulls. Inf. Process. Lett. 9, 223–228 (1979)

    Article  MATH  Google Scholar 

  7. Bürgisser, P., Clausen, M., Shokrollahi, M.A.: Algebraic Complexity Theory. Springer, Berlin (1997)

  8. Edelsbrunner, H.: Geometry and Topology for Mesh Generation (1st paperback ed.). Cambridge University Press, Cambridge (2006)

  9. Futer, D., Guéritaud, F.: From angled triangulations to hyperbolic structures. In: Champanerkar, A. (ed.) Interactions Between Hyperbolic Geometry, Quantum Topology and Number Theory. Contemporary Mathematics, vol. 541, pp. 159–182. American Mathematical Society, Providence, RI (2011)

  10. Fortune, S.: Voronoi diagrams and Delaunay triangulations. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry. Discrete Mathematics and Its Applications, 2nd edn, pp. 513–528. Chapman & Hall/CRC, Boca Raton (2004)

  11. Garling, D.J.H.: Inequalities: A Journey into Linear Analysis. Cambridge University Press, Cambridge (2007)

    Book  Google Scholar 

  12. Gonska, B., Padrol, A.: Neighborly inscribed polytopes and Delaunay triangulations. Adv. Geom. (to appear). arXiv:1308.5798

  13. Gonska, B., Ziegler, G.M.: Inscribable stacked polytopes. Adv. Geom. 13(4), 723–740 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  14. Goodman, J.E., O’Rourke, J.: Handbook of Discrete and Computational Geometry. Discrete Mathematics and Its Applications, 2nd edn. Chapman & Hall/CRC, Boca Raton (2004)

  15. Henk, M., Richter-Gebert, J., Ziegler, G.M.: Basic properties of convex polytopes. In: Handbook of Discrete and Computational Geometry. Discrete Mathematics and Its Applications, 2nd edn, pp. 355–382. Chapman & Hall/CRC, Boca Raton (2004)

  16. Jaggi, B., Mani-Levitska, P., Sturmfels, B., White, N.: Uniform oriented matroids without the isotopy property. Discrete Comput. Geom. 4(2), 97–100 (1989)

    MathSciNet  Article  MATH  Google Scholar 

  17. Kortenkamp, U.H.: Every simplicial polytope with at most \(d+4\) vertices is a quotient of a neighborly polytope. Discrete Comput. Geom. 18(4), 455–462 (1997)

    MathSciNet  Article  MATH  Google Scholar 

  18. Mnëv, N.E.: The universality theorems on the classification problem of configuration varieties and convex polytopes varieties. Topology and Geometry–Rohlin Seminar. Lecture Notes in Mathematics, vol. 1346, pp. 527–544. Springer, Heidelberg (1988)

  19. Munkres, J.R.: Topology: A First Course. Prentice-Hall Inc, Englewood Cliffs (1975)

    MATH  Google Scholar 

  20. Padrol, A.: Many neighborly polytopes and oriented matroids. Discrete Comput. Geom. 50(4), 865–902 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  21. Padrol, A., Theran, L.: Delaunay triangulations with disconnected realization spaces. In: Cheng S.-W., Devillers O. (eds) Symposium on Computational Geometry, p. 163. ACM, New York (2014)

  22. Richter-Gebert, J.: Mnëv’s universality theorem revisited. Sém. Lothar. Combin. 34 (1995), Art. B34h, approx. 15 pp (electronic)

  23. Richter-Gebert, J.: Realization Spaces of Polytopes. Lecture Notes in Mathematics, vol. 1643. Springer, Berlin (1996)

  24. Richter-Gebert, J.: Two interesting oriented matroids. Doc. Math. J. DMV 1, 137–148 (1996)

    MathSciNet  MATH  Google Scholar 

  25. Richter-Gebert, J.: The universality theorems for oriented matroids and polytopes, In: Chazelle, B., Goodman, J.E., Pollack, R. (eds.) Advances in Discrete and Computational Geometry (Mount Holyoke 1996). Contemporary Mathematics, vol. 223, pp. 269–292. American Mathematics Society, Providence, RI (1998)

  26. Richter-Gebert, J., Ziegler, G.M.: Oriented matroids. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry. Discrete Mathematics and Its Applications, 2nd edn, pp. 129–151. Chapman & Hall/CRC, Boca Raton (2004)

  27. Rivin, I.: Euclidean structures on simplicial surfaces and hyperbolic volume. Ann. Math. (2) 139(3), 553–580 (1994)

    MathSciNet  Article  Google Scholar 

  28. Rivin, I.: A characterization of ideal polyhedra in hyperbolic 3-space. Ann. Math. (2) 143(1), 51–70 (1996)

    MathSciNet  Article  MATH  Google Scholar 

  29. Rivin, I.: Combinatorial optimization in geometry. Adv. Appl. Math. 31(1), 242–271 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  30. Seidel, R.: A method for proving lower bounds for certain geometric problems. In: Toussaint, G.T. (ed.) Computational Geometry, pp. 319–334. Springer, North-Holland (1985)

    Google Scholar 

  31. Shemer, I.: Neighborly polytopes. Isr. J. Math. 43, 291–314 (1982)

    MathSciNet  Article  MATH  Google Scholar 

  32. Shor, P.W.: Stretchability of pseudolines is NP-hard. In: Bezdek, A., Kuperberg, W. (eds.) Applied Geometry and Discrete Mathematics. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 4, pp. 531–554. American Mathematical Society, Providence, RI (1991)

  33. Steiner, J.: Systematische Entwicklung der Abhängigkeit geometrischer Gestalten von einander. Fincke, Berlin, 1832, Also. In: Gesammelte Werke, vol. 1, pp. 229–458. Reimer, Berlin (1881)

  34. Steinitz, E.: Über isoperimetrische Probleme bei konvexen Polyedern. J. Reine Angew. Math. 159, 133–143 (1928)

  35. Sturmfels, B.: Neighborly polytopes and oriented matroids. Eur. J. Comb. 9(6), 537–546 (1988)

    MathSciNet  Article  MATH  Google Scholar 

  36. Suvorov, P.: Isotopic but not rigidly isotopic plane systems of straight lines. Topology and Geometry–Rohlin Seminar. Lecture Notes in Mathematics, vol. 1346, pp. 545–556. Springer, Heidelberg (1988)

  37. Tsukamoto, Y.: New examples of oriented matroids with disconnected realization spaces. Discrete Comput. Geom. 49(2), 287–295 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  38. Vakil, R.: Murphy’s law in algebraic geometry: badly-behaved deformation spaces. Invent. Math. 164(3), 569–590 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  39. Valiant, L.G.: Completeness classes in algebra. In: Conference Record of the Eleventh Annual ACM Symposium on Theory of Computing, pp. 249–261. ACM, New York (1979)

  40. Vershik, A.M.: Topology of the convex polytopes’ manifolds, the manifold of the projective configurations of a given combinatorial type and representations of lattices. Topology and Geometry–Rohlin Seminar. Lecture Notes in Mathematics, vol. 1346, pp. 557–581. Springer, Heidelberg (1988)

  41. White, N.L.: A nonuniform matroid which violates the isotopy conjecture. Discrete Comput. Geom. 4(1), 1–2 (1989)

    MathSciNet  Article  MATH  Google Scholar 

  42. Ziegler, G.M.: Lectures on Polytopes. Graduate Texts in Mathematics, vol. 152. Springer, New York (1995)

    Book  Google Scholar 

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Acknowledgments

K.A. Adiprasito acknowledges the support by an EPDI postdoctoral fellowship and the support by a Minerva Fellowship from the Max Planck Society and by the Romanian NASR, CNCS—UEFISCDI, project PN-II-ID-PCE-2011-3-0533. The research of A. Padrol is supported by the DFG Collaborative Research Center SFB/TR 109 “Discretization in Geometry and Dynamics.” The research of L. Theran was carried out at Inst. Math., Freie Universität Berlin and was supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement No. 247029-SDModels. A preliminary version of some of the results of this paper has appeared in [21].

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Correspondence to Arnau Padrol.

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Adiprasito, K.A., Padrol, A. & Theran, L. Universality Theorems for Inscribed Polytopes and Delaunay Triangulations. Discrete Comput Geom 54, 412–431 (2015). https://doi.org/10.1007/s00454-015-9714-x

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  • DOI: https://doi.org/10.1007/s00454-015-9714-x

Keywords

  • Inscribed polytope
  • Delaunay triangulation
  • Realization space
  • Universality theorem