Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2016, Article ID 4873276, 17 pages
http://dx.doi.org/10.1155/2016/4873276
Research Article

Bounding Regions to Plane Steepest Descent Curves of Quasiconvex Families

1Dipartimento di Matematica e Informatica Ulisse Dini, Università degli Studi di Firenze, Viale Morgagni 67/a, 50134 Firenze, Italy
2Dipartimento GESAAF, Università degli Studi di Firenze, Piazzale delle Cascine 15, 50144 Firenze, Italy

Received 26 February 2016; Accepted 17 April 2016

Academic Editor: Wenyu Sun

Copyright © 2016 Marco Longinetti et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. W. Fenchel, Convex Cones, Sets and Functions, Princeton University Press, Princeton, NJ, USA, 1953.
  2. C. Icking, R. Klein, and E. Langetepe, “Self-approaching curves,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 125, no. 3, pp. 441–453, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. I. F. Maǐnik, “An estimate of length for the curve of descent,” Sibirskiĭ Matematicheskiĭ Zhurnal, vol. 33, no. 4, pp. 215–218, 1992. View at Google Scholar
  4. P. Manselli and C. Pucci, “Maximum length of steepest descent curves for quasi-convex functions,” Geometriae Dedicata, vol. 38, no. 2, pp. 211–227, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  5. J. Bolte, A. Daniilidis, O. Ley, and L. Mazet, “Characterizations of Łojasiewicz inequalities: subgradient flows, talweg, convexity,” Transactions of the American Mathematical Society, vol. 362, no. 6, pp. 3319–3363, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. A. Daniilidis, G. David, E. Durand-Cartagena, and A. Lemenant, “Rectifiability of self-contracted curves in the euclidean space and applications,” Journal of Geometric Analysis, vol. 25, no. 2, pp. 1211–1239, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. A. Daniilidis, O. Ley, and S. Sabourau, “Asymptotic behaviour of self-contracted planar curves and gradient orbits of convex functions,” Journal de Mathématiques Pures et Appliquées, vol. 94, no. 2, pp. 183–199, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. M. Longinetti, P. Manselli, and A. Venturi, “On steepest descent curves for quasi convex families in Rn,” Mathematische Nachrichten, vol. 288, no. 4, pp. 420–442, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. M. Longinetti, P. Manselli, and A. Venturi, “On variational problems related to steepest descent curves and self-dual convex sets on the sphere,” Applicable Analysis, vol. 94, no. 2, pp. 294–307, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. T. Bonnesen and W. Fenchel, Theory of Convex Bodies, BCS Associates, 1987. View at MathSciNet
  11. R. Schneider, Convex Bodies: the Brunn-Minkowski Theory, vol. 44, Cambridge University Press, Cambridge, UK, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  12. B. de Finetti, “Sulle stratificazioni convesse,” Annali di Matematica Pura ed Applicata, vol. 30, no. 4, pp. 173–183, 1949. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. H. W. Guggenheimer, Differential Geometry, McGraw-Hill, 1963. View at MathSciNet
  14. R. T. Rockafellar, Convex Analysis, vol. 28 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, USA, 1970.