On steepest descent curves for quasi convex families in R^n

A connected, linearly ordered subset of an euclidean space satisfying the property that the distance of any previous points is not decreasing with repect to the order 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.