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

TL;DR

This paper investigates properties of steepest descent curves within quasi convex families in R^n, establishing their rectifiability, bounds on length, and their relation to quasi convex functions, with proofs of existence, uniqueness, and stability.

Contribution

It introduces generalized steepest descent curves for quasi convex functions, proving their fundamental properties and establishing bounds and stability results.

Findings

Steepest descent curves are rectifiable with bounded length.

Existence, uniqueness, and stability of these curves are established.

Generalizations of steepest descent curves for quasi convex functions are introduced.

Abstract

A connected, linearly ordered path $\ga \subset R^n$ satisfying $$ x_1\prec x_2\prec x_3 \in \ga, and x_1 \prec x_2 \prec x_3 \Rightarrow |x_2 - x_1| \leq | x_3 - x_1|$$ 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.