On steepest descent curves for quasi convex families in Rn
Marco Longinetti, Paolo Manselli, Adriana Venturi · Mathematische Nachrichten · 2014
A connected, linearly ordered subset γ ⊂ R n satisfying x 1 , x 2 , x 3 ∈ γ , and x 1 ≺ x 2 ≺ x 3 → | x 2 − x 1 | ≤ | x 3 − x 1 | 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.