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