On steepest descent curves for quasi convex families in Rn

A connected, linearly ordered subset gamma subset of R-n satisfying x(1), x(2), x(3) epsilon gamma, and x(1) < x(2) < x(3) double right arrow is shown to be a rectifiable curve; a priori bounds for its length are given; moreover, these curves are generalized steepest descent curves of suitable quasi convex functions. Properties of quasi convex families are considered; special curves related to quasi convex families are defined and studied; they are generalizations of steepest descent curves for quasi convex functions and satisfy the previous property. Existence, uniqueness, stability results and length bounds are proved for them. (C) 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim