Skip to main content
SpringerLink
Log in

The universality theorem for neighborly polytopes

  • Original Paper
  • Published:
Combinatorica Aims and scope Submit manuscript

Abstract

In this note, we prove that every open primary basic semialgebraic set is stably equivalent to the realization space of a neighborly simplicial polytope. This in particular provides the final step for Mnëv‘s proof of the universality theorem for simplicial polytopes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. Altshuler and L. Steinberg: Neighborly 4-polytopes with 9 vertices, J. Combinatorial Theory Ser. A 15 (1973), 270–287.

    Article  MathSciNet  MATH  Google Scholar 

  2. A. Björner, M. Las Vergnas, B. Sturmfels, N. White, and G. M. Ziegler: Oriented matroids, second ed., Encyclopedia of Mathematics and its Applications, vol. 46, Cambridge University Press, Cambridge, 1999.

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  4. J. Bokowski and B. Sturmfels: On the coordinatization of oriented matroids, Discrete Comput. Geom. 1 (1986), 293–306.

    Article  MathSciNet  MATH  Google Scholar 

  5. H. Günzel: The universal partition theorem for oriented matroids, Discrete Comput. Geom. 15 (1996), 121–145.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. N. E. Mnëv: On manifolds of combinatorial types of projective configurations and convex polyhedra, Sov. Math., Dokl. 32 (1985), 335–337.

    MATH  Google Scholar 

  9. N. E. Mnëv: The topology of configuration varieties and convex polytopes varieties, Ph.D. thesis, St. Petersburg State University, St. Petersburg, RU, 1986, 116 pages, pdmi.ras.ru/~mnev/mnev phd1.pdf.

    MATH  Google Scholar 

  10. N. E. Mnëv: The universality theorems on the classification problem of configuration varieties and convex polytopes varieties, Topology and geometry, Rohlin Semin. 1984-1986, Lect. Notes Math. 1346, 527–543, 1988.

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  14. J. Richter-Gebert and G. M. Ziegler: Realization spaces of 4-polytopes are universal, Bulletin of the American Mathematical Society 32 (1995), 403–412.

    Article  MathSciNet  MATH  Google Scholar 

  15. J. Richter-Gebert: The universality theorems for oriented matroids and poly-topes, Advances in discrete and computational geometry (South Hadley, MA, 1996), Contemp. Math., vol. 223, Amer. Math. Soc., Providence, RI, 1999, 269–292.

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. P. W. Shor: Stretchability of pseudolines is NP-hard, Applied Geometry and Discrete Mathematics — The Victor Klee Festschrift (P. Gritzmann and B. Sturmfels, eds.), DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 4, Amer. Math. Soc., Providence RI, 1991, 531–554.

    Google Scholar 

  18. B. Sturmfels: Neighborly polytopes and oriented matroids, European J. Combin. 9 (1988), 537–546.

    Article  MathSciNet  MATH  Google Scholar 

  19. B. Sturmfels: Simplicial polytopes without the isotopy property, preprints of the Institute for Mathematics and Applications (1988), 5.

    Google Scholar 

  20. G. M. Ziegler: Lectures on Polytopes, Graduate Texts in Mathematics, vol. 152, Springer, New York, 1995, Revised edition, 1998; seventh updated printing 2007.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Einstein Institute for Mathematics, Hebrew University of Jerusalem, Jerusalem, Israel

    Karim A. Adiprasito

  2. Institut de Mathématiques de Jussieu - Paris Rive Gauche, Universit Pierre et Marie Curie (Paris 06), Paris, France

    Arnau Padrol

Authors
  1. Karim A. Adiprasito
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Arnau Padrol
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Karim A. Adiprasito.

Additional information

K. A. Adiprasito acknowledges support by an EPDI postdoctoral fellowship and by the Romanian NASR, CNCS — UEFISCDI, project PN-II-ID-PCE-2011-3-0533. The research of A. Padrol was supported by the DFG Collaborative Research Center SFB/TR 109 “Discretization in Geometry and Dynamics”.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Adiprasito, K.A., Padrol, A. The universality theorem for neighborly polytopes. Combinatorica 37, 129–136 (2017). https://doi.org/10.1007/s00493-016-3253-9

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00493-016-3253-9

Mathematics Subject Classification (2000)

Search

Navigation