Skip to main content
SpringerLink
Log in

Optimal Inequalities Between Distances in Convex Projective Domains

  • Published:
The Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

On any proper convex domain in real projective space there exists a natural Riemannian metric, the Blaschke metric. On the other hand, distances between points can be measured in the Hilbert metric. Using techniques of optimal control, we develop inequalities that provide lower bounds for the Riemannian length of the line segment joining two points of the domain by the Hilbert distance between these points. This strengthens a result of Tholozan. Our estimates are valid for a whole class of Riemannian metrics on convex projective domains, namely those induced by convex non-degenerate centro-affine hypersurface immersions. If the immersions are asymptotic to the boundary of the convex cone over the domain, then we can also upper bound the Riemmanian length. On these classes, and in particular for the Blaschke metric, our inequalities are optimal.

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
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Bellman, R.: Dynamic Programming. Dover, New York (2003)

    MATH  Google Scholar 

  2. Benoist, Y., Hulin, D.: Cubic differentials and finite volume convex projective surfaces. Geom. Topol. 17(1), 595–620 (2013)

    Article  MathSciNet  Google Scholar 

  3. Benoist, Y., Hulin, D.: Cubic differentials and hyperbolic convex sets. J. Differ. Geom. 98(1), 1–19 (2014)

    Article  MathSciNet  Google Scholar 

  4. Benzécri, J.P.: Sur les variétés localement affines et localement projectives. Bull. Soc. Math. France 88, 229–332 (1960)

    Article  MathSciNet  Google Scholar 

  5. Gelfand, I., Fomin, S.: Calculus of Variations. Dover, New York (1963)

    MATH  Google Scholar 

  6. Hildebrand, R.: On the infinity-norm of the cubic form of complete hyperbolic affine hyperspheres. Results Math. 64(1), 113–119 (2013)

    Article  MathSciNet  Google Scholar 

  7. Kim, I., Papadopoulos, A.: Convex real projective structures and Hilbert metrics. In: Hand-Book of Hilbert Geometry, pp. 307–338. European Mathematical Society Publishing House, Zurich (2014)

  8. Loftin, J.C.: Affine spheres and convex \(\mathbb{RP}^n\)-manifolds. Am. J. Math. 123(2), 255–275 (2001)

    Article  Google Scholar 

  9. Loftin, J., MacIntosh, I.: Cubic differentials in the differential geometry of surfaces. In: Handbook of Teichmüller theory V. European Mathematical Society, Zürich (2015)

  10. Nomizu, K., Sasaki, T.: Affine Differential Geometry: Geometry of Affine Immersions. Cambridge Tracts in Mathematics, vol. 111. Cambridge University Press, Cambridge (1994)

  11. Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mischchenko, E.F.: The Mathematical Theory of Optimal Processes. Wiley, New York (1962)

    Google Scholar 

  12. Socié-Méthou, E.: Behaviour of distance functions in Hilbert–Finsler geometry. Differ. Geom. Appl. 20(1), 1–10 (2004)

    Article  MathSciNet  Google Scholar 

  13. Tholozan, N.: Volume entropy of Hilbert metrics and length spectrum of Hitchin representations into PSL(3, R). Duke Math. J. 166(7), 1377–1403 (2017)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University Grenoble Alpes, CNRS, Grenoble INP, LJK, 38000, Grenoble, France

    Roland Hildebrand

Authors
  1. Roland Hildebrand
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Roland Hildebrand.

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

Hildebrand, R. Optimal Inequalities Between Distances in Convex Projective Domains. J Geom Anal 31, 11357–11385 (2021). https://doi.org/10.1007/s12220-021-00684-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12220-021-00684-3

Keywords

Mathematics Subject Classification

Search

Navigation