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Journal of Geometry

, 110:40 | Cite as

Radial expansion preserves hyperbolic convexity and radial contraction preserves spherical convexity

  • Dhruv KohliEmail author
  • Jeffrey M. Rabin
Article

Abstract

On a flat plane, convexity of a set is preserved by both radial expansion and contraction of the set about any point inside it. Using the Poincaré disk model of hyperbolic geometry, we prove that radial expansion of a hyperbolic convex set about a point inside it always preserves hyperbolic convexity. Using stereographic projection of a sphere, we prove that radial contraction of a spherical convex set about a point inside it, such that the initial set is contained in the closed hemisphere centred at that point, always preserves spherical convexity.

Keywords

Preserving hyperbolic and spherical convexity Poincaré disk stereographic projection dilation radial expansion and contraction 

Notes

Acknowledgements

We would like to thank M. Xiao for several useful discussions that helped in defining scaling transformations in the Poincaré disk. We also thank the anonymous referee for generous comments that greatly improved this paper.

Compliance with ethical standards

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

References

  1. 1.
    Beardon, A.F.: The Geometry of Discrete Groups. Springer, New York (1982)Google Scholar
  2. 2.
    Beardon, A.F., Minda, D.: The hyperbolic metric and geometric function theory. In: Ponnusamy, S., Sugawa, T., Vuorinen, M. (eds.) Quasiconformal Mappings and their Applications, pp. 9–56. Narosa Publishing House, New Delhi (2007)zbMATHGoogle Scholar
  3. 3.
    Ma, W., Minda, D.: Geometric properties of hyperbolic geodesics. In : Proceedings of the International Workshop on Quasiconformal Mappings and their Applications (2007)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California, San DiegoLa JollaUSA

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