Discrete & Computational Geometry

, Volume 54, Issue 2, pp 412–431 | Cite as

Universality Theorems for Inscribed Polytopes and Delaunay Triangulations

  • Karim A. Adiprasito
  • Arnau Padrol
  • Louis Theran


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.


Inscribed polytope Delaunay triangulation Realization space Universality theorem 



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].


  1. 1.
    Adiprasito, K.A., Padrol, A.: The universality theorem for neighborly polytopes. Combinatorica (to appear). arXiv:1402.7207
  2. 2.
    Adiprasito, K.A., Ziegler, G.M.: Many projectively unique polytopes. Invent. Math. 199(3), 581–652 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 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. 4.
    Bokowski, J., Guedes de Oliveira, A.: Simplicial convex 4-polytopes do not have the isotopy property. Port. Math. 47(3), 309–318 (1990)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bokowski, J., Ewald, G., Kleinschmidt, P.: On combinatorial and affine automorphisms of polytopes. Isr. J. Math. 47(2–3), 123–130 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Brown, K.Q.: Voronoi diagrams from convex hulls. Inf. Process. Lett. 9, 223–228 (1979)CrossRefzbMATHGoogle Scholar
  7. 7.
    Bürgisser, P., Clausen, M., Shokrollahi, M.A.: Algebraic Complexity Theory. Springer, Berlin (1997)Google Scholar
  8. 8.
    Edelsbrunner, H.: Geometry and Topology for Mesh Generation (1st paperback ed.). Cambridge University Press, Cambridge (2006)Google Scholar
  9. 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)Google Scholar
  10. 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)Google Scholar
  11. 11.
    Garling, D.J.H.: Inequalities: A Journey into Linear Analysis. Cambridge University Press, Cambridge (2007)CrossRefGoogle Scholar
  12. 12.
    Gonska, B., Padrol, A.: Neighborly inscribed polytopes and Delaunay triangulations. Adv. Geom. (to appear). arXiv:1308.5798
  13. 13.
    Gonska, B., Ziegler, G.M.: Inscribable stacked polytopes. Adv. Geom. 13(4), 723–740 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 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)Google Scholar
  15. 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)Google Scholar
  16. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 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)Google Scholar
  19. 19.
    Munkres, J.R.: Topology: A First Course. Prentice-Hall Inc, Englewood Cliffs (1975)zbMATHGoogle Scholar
  20. 20.
    Padrol, A.: Many neighborly polytopes and oriented matroids. Discrete Comput. Geom. 50(4), 865–902 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 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)Google Scholar
  22. 22.
    Richter-Gebert, J.: Mnëv’s universality theorem revisited. Sém. Lothar. Combin. 34 (1995), Art. B34h, approx. 15 pp (electronic)Google Scholar
  23. 23.
    Richter-Gebert, J.: Realization Spaces of Polytopes. Lecture Notes in Mathematics, vol. 1643. Springer, Berlin (1996)Google Scholar
  24. 24.
    Richter-Gebert, J.: Two interesting oriented matroids. Doc. Math. J. DMV 1, 137–148 (1996)MathSciNetzbMATHGoogle Scholar
  25. 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)Google Scholar
  26. 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)Google Scholar
  27. 27.
    Rivin, I.: Euclidean structures on simplicial surfaces and hyperbolic volume. Ann. Math. (2) 139(3), 553–580 (1994)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Rivin, I.: A characterization of ideal polyhedra in hyperbolic 3-space. Ann. Math. (2) 143(1), 51–70 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Rivin, I.: Combinatorial optimization in geometry. Adv. Appl. Math. 31(1), 242–271 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 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. 31.
    Shemer, I.: Neighborly polytopes. Isr. J. Math. 43, 291–314 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 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)Google Scholar
  33. 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)Google Scholar
  34. 34.
    Steinitz, E.: Über isoperimetrische Probleme bei konvexen Polyedern. J. Reine Angew. Math. 159, 133–143 (1928)Google Scholar
  35. 35.
    Sturmfels, B.: Neighborly polytopes and oriented matroids. Eur. J. Comb. 9(6), 537–546 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 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)Google Scholar
  37. 37.
    Tsukamoto, Y.: New examples of oriented matroids with disconnected realization spaces. Discrete Comput. Geom. 49(2), 287–295 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Vakil, R.: Murphy’s law in algebraic geometry: badly-behaved deformation spaces. Invent. Math. 164(3), 569–590 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 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)Google Scholar
  40. 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)Google Scholar
  41. 41.
    White, N.L.: A nonuniform matroid which violates the isotopy conjecture. Discrete Comput. Geom. 4(1), 1–2 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Ziegler, G.M.: Lectures on Polytopes. Graduate Texts in Mathematics, vol. 152. Springer, New York (1995)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Karim A. Adiprasito
    • 1
  • Arnau Padrol
    • 2
  • Louis Theran
    • 3
  1. 1.Einstein Institute of MathematicsHebrew University of JerusalemJerusalemIsrael
  2. 2.Institut für MathematikFreie Universität BerlinBerlinGermany
  3. 3.Aalto Science Institute and Department of Computer ScienceAalto UniversityAaltoFinland

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