Geometriae Dedicata

, Volume 38, Issue 2, pp 211–227 | Cite as

Maximum length of steepest descent curves for quasi-convex functions

  • Paolo Manselli
  • Carlo Pucci


Plane, oriented, rectifiable curves, such that, in almost each x the normal line bounds a half-plane containing the part of the curve preceding x, are considered. It is shown that, in the family of curves as above, with convex hull of given perimeter, there exist curves of maximal length and these are evolutes of themselves. As a consequence, it is proved that quasi-convex functions in a set Ω have steepest descent lines with length bounded by the diameter of Ω. This result is then extended to ℝ n .


Convex Hull Maximal Length Steep Descent Normal Line Descent Line 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    De Finetti, B., ‘Sulle stratificazioni convesse’, Ann. Mat. Pura Appl. (4) 30 (1949), 173–183.Google Scholar
  2. 2.
    Eggleston, H. G., Convexity, Cambridge Univ. Press, 1958.Google Scholar
  3. 3.
    Fenchel, W., Convex Cones, Sets, and Functions, Princeton Univ. Press, 1953.Google Scholar
  4. 4.
    Guggenheimer, H. W., Differential Geometry, Dover, New York, 1977.Google Scholar
  5. 5.
    Santalò, L. A., ‘Convex regions on the n-dimensional spherical surface’, Ann. of Math. (2) 47 (1946), 448–459.Google Scholar
  6. 6.
    Santalò, L. A., Integral Geometry and Geometric Probability, Addison Wesley, 1976.Google Scholar

Copyright information

© Kluwer Academic Publishers 1991

Authors and Affiliations

  • Paolo Manselli
    • 1
  • Carlo Pucci
    • 2
  1. 1.Istituto di Matematica, Facoltà di ArchitetturaUniversità di FirenzeFirenzeItaly
  2. 2.Istituto di Analisi Globale del C.N.R.FirenzeItaly

Personalised recommendations