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

A connected, linearly ordered path $\text{ga}\subset R^n$ satisfying $$x_1\prec x_2\prec x_3\in \text{ga},and{x}_1\prec x_2\prec x_3\Rightarrow \mathrm{\mid }{x}_2-x_1\mathrm{\mid }\le \mathrm{\mid }{x}_3-x_1\mathrm{\mid }$$ is shown to be a rectifiable curve; a priori bounds for its length are given; moreover, these paths 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’s bounds are proved for them.