Asymptotic behaviour of self-contracted planar curves and gradient orbits of convex functions

TL;DR

This paper introduces self-contracted curves, proves bounded planar curves have finite length, and explores gradient orbits of convex functions, including examples of spiraling orbits around minima.

Contribution

It defines self-contracted curves, establishes their finite length property in the plane, and analyzes gradient orbits, including a spiraling example for convex functions.

Findings

Bounded self-contracted planar curves have finite length.

Gradient orbits of convex functions can spiral infinitely around minima.

The paper provides an example of a spiraling gradient orbit in the plane.

Abstract

We hereby introduce and study the notion of self-contracted curves, which encompasses orbits of gradient systems of convex and quasiconvex functions. Our main result shows that bounded self-contracted planar curves have a finite length. We also give an example of a convex function defined in the plane whose gradient orbits spiral infinitely many times around the unique minimum of the function.