Maximum length of steepest descent curves for quasi-convex functions

Abstract

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 ℝⁿ.