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Positivity

, Volume 18, Issue 3, pp 505–517 | Cite as

Lipschitz and Hölder continuity results for some functions of cones

  • Alberto SeegerEmail author
Article

Abstract

We prove the Lipschitz continuity of the maximal angle function on the set of closed convex cones in a Hilbert space. A similar result is obtained for the minimal angle function. On the other hand, we prove that the incenter of a solid cone and the circumcenter of a sharp cone behave in a locally Hölderian manner.

Keywords

Convex cone Maximal angle Incenter Circumcenter Solidity coefficient Sharpness coefficient  Lipschitz continuity Hölder continuity 

Mathematics Subject Classification (2000)

26A12 26B35 56A55 52A40 90C25 90C26 

Notes

Acknowledgments

The author would like to thank the referee for meticulous reading of the manuscript and for several suggestions that improved the presentation.

References

  1. 1.
    Ahn, H.K., Bae, S.W., Cheong, O., Gudmundsson, J.: Aperture-angle and Hausdorff approximation of convex figures. Discrete Comput. Geom. 40, 414–429 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Astorino, A., Gaudioso, M., Seeger, A.: An illumination problem: optimal apex and optimal orientation for a cone of light. J. Global Optim. online April 2013. doi:  10.1007/s10898-013-0071-0
  3. 3.
    Attouch, H., Wets, R.J.-B.: Quantitative stability of variational systems. II. A framework for nonlinear conditioning. SIAM J. Optim. 3, 359–381 (1993)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Bose, P., Hurtado-Diaz, F., Omana-Pulido, E., Snoeyink, J., Toussaint, G.T.: Some aperture-angle optimization problems. Algorithmica 33, 411–435 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Clarke, F.H.: Optimization and nonsmooth analysis. In: Canadian Mathematical Society Series of Monographs and Advanced Texts. A Wiley-Interscience Publication, New York (1983)Google Scholar
  6. 6.
    Dunkl, C.F., Williams, K.S.: A simple norm inequality. Am. Math. Monthly 71, 53–54 (1964)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Freund, R., Vera, J.R.: Condition-based complexity of convex optimization in conic linear form via the ellipsoid algorithm. SIAM J. Optim. 10, 155–176 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Gourion, D., Seeger, A.: Solidity indices for convex cones. Positivity 16, 685–705 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Henrion, R., Seeger, A.: On properties of different notions of centers for convex cones. Set-Valued Var. Anal. 18, 205–231 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Henrion, R., Seeger, A.: Inradius and circumradius of various convex cones arising in applications. Set-Valued Var. Anal. 18, 483–511 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Henrion, R., Seeger, A.: Condition number and eccentricity of a closed convex cone. Math. Scand. 109, 285–308 (2011)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Iusem, A., Seeger, A.: Axiomatization of the index of pointedness for closed convex cones. Comput. Appl. Math. 24, 245–283 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Iusem, A., Seeger, A.: On pairs of vectors achieving the maximal angle of a convex cone. Math. Program. Ser. B. 104, 501–523 (2005)Google Scholar
  14. 14.
    Iusem, A., Seeger, A.: Normality and modulability indices. I. Convex cones in normed spaces. J. Math. Anal. Appl. 338, 365–391 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Iusem, A., Seeger, A.: Normality and modulability indices. II. Convex cones in Hilbert spaces. J. Math. Anal. Appl. 338, 392–406 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Iusem, A., Seeger, A.: Searching for critical angles in a convex cone. Math. Program. Ser. B. 120, 3–25 (2009)Google Scholar
  17. 17.
    Jameson, G.: Ordered linear spaces. In: Lecture Notes in Mathematics, vol. 141. Springer, Berlin (1970)Google Scholar
  18. 18.
    Krasnosel’skij, M.A., Lifshits, Je. A., Sobolev, A.V.: Positive linear systems. In: The Method of Positive Operators. Heldermann Verlag, Berlin (1989)Google Scholar
  19. 19.
    Omana-Pulido, E., Toussaint, G.T.: Aperture-angle optimization problems in three dimensions. J. Math. Model. Algorithms 1, 301–329 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Pena, J., Renegar, J.: Computing approximate solutions for convex conic systems of constraints. Math. Program. Ser. A 87, 351–383 (2000)Google Scholar
  21. 21.
    Sitarz, S.: The medal points’ incenter for rankings in sport. Appl. Math. Lett. 26, 408–412 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Walkup, D.W., Wets, R.J.B.: Continuity of some convex-cone-valued mappings. Proc. Am. Math. Soc. 18, 229–235 (1967)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of AvignonAvignonFrance

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