On steepest descent curves for quasi convex families in Rⁿ

2013/03/15 by Marco Longinetti, Longinetti, Marco, Paolo Manselli +3
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP

paper · pdf · doi:10.48550/arxiv.1303.3721

29 pages, 1 figure

arxiv created 2013/03/15 · arxiv updated 2013/03/18

Abstract

A connected, linearly ordered path \ga ⊂ Rⁿ satisfying x₁\prec x₂\prec x₃ ∈ \ga, and x₁ \prec x₂ \prec x₃ ⇒ |x₂ - x₁| ≤ | x₃ - x₁| 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.

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