Skip to main content
SpringerLink
Log in

The Hilbert area of inscribed triangles and quadrilaterals

  • Original Paper
  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

Hilbert volume is an invariant of real projective geometry. Polygons inscribed in polygons are considered for the real projective plane. In two-dimensions the correspondence between Fock–Goncharov and Cartesian coordinates is examined. Degeneration and Hilbert area of inscribed quadrilaterals are analyzed. A microlocal condition is developed for bounded Hilbert area under degeneration. The condition is applied to give a sequence of strictly convex domains with bounded Hilbert area and divergent Goldman parameters.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Benoist, Y.: Convexes divisibles. I. In: Algebraic Groups and Arithmetic, pp. 339–374. Tata Institute of Fundamental Research, Mumbai, (2004)

  2. Bonahon, F., Kim, I.: The Goldman and Fock-Goncharov coordinates for convex projective structures on surfaces. Geom. Dedicata 192, 43–55 (2018)

    Article  MathSciNet  Google Scholar 

  3. Casella, A., Tate, D., Tillmann, S.: Moduli spaces of real projective structures on surfaces: Notes on a paper by V.V. Fock and A.B. Goncharov. arXiv:1801.03913 (math) (2018)

  4. Colbois, B., Vernicos, C., Verovic, P.: L’aire des triangles idéaux en géométrie de Hilbert. Enseign. Math. (2) 50(3–4), 203–237 (2004)

    MathSciNet  MATH  Google Scholar 

  5. Colbois, B., Vernicos, C., Verovic, P.: Area of ideal triangles and Gromov hyperbolicity in Hilbert geometry. Illinois J. Math. 52(1), 319–343 (2008)

    Article  MathSciNet  Google Scholar 

  6. Fock, V.V., Goncharov, A.B.: Moduli spaces of convex projective structures on surfaces. Adv. Math. 208(1), 249–273 (2007)

    Article  MathSciNet  Google Scholar 

  7. Foulon, P., Kim, I.: Topological entropy and bulging deformation of real projective structures on surfaces. arXiv:1608.06799 (math) (2016)

  8. Goldman, W.M.: Convex real projective structures on compact surfaces. J. Differ. Geom. 31(3), 791–845 (1990)

    Article  MathSciNet  Google Scholar 

  9. Ilesanmi, A., Daryl, C.: The area of convex projective surfaces and Fock-Goncharov coordinates. arXiv:1506.08245 (math) (2015)

  10. Kim, I.: Degeneration of strictly convex real projective structures on surface. arXiv:1811.11841 (math) (2018)

  11. Nie, X.: On the Hilbert geometry of simplicial Tits sets. Ann. Inst. Fourier (Grenoble) 65(3), 1005–1030 (2015)

    Article  MathSciNet  Google Scholar 

  12. Sun, Z.: Volume of the moduli space of unmarked bounded positive convex \({{\mathbb{R}}}{{\mathbb{P}}}^2\) structures. arXiv:2001.01295 (math) (2020)

  13. Zhang, T.: The degeneration of convex \({\mathbb{R}}{\mathbb{P}}^2\) structures on surfaces. Proc. Lond. Math. Soc. (3) 111(5), 967–1012 (2015)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Maryland, College Park, MD, 20742, USA

    Scott A. Wolpert

Authors
  1. Scott A. Wolpert
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Scott A. Wolpert.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wolpert, S.A. The Hilbert area of inscribed triangles and quadrilaterals. Geom Dedicata 214, 177–192 (2021). https://doi.org/10.1007/s10711-021-00610-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-021-00610-5

Keywords

Mathematics Subject Classification (2010)

Search

Navigation