Rectifiability of non Euclidean planar self-contracted curves

TL;DR

This paper proves that self-contracted curves in the plane with a smooth, strictly convex norm have finite length, extending understanding of their geometric properties in non-Euclidean settings.

Contribution

It establishes the finite length property of self-contracted curves in non-Euclidean planes with smooth, strictly convex norms, generalizing previous Euclidean results.

Findings

Self-contracted curves in R^2 with a C^2, strictly convex norm have finite length.

The proof involves analyzing the curve bisector in the general norm setting.

The approach adapts techniques from Euclidean geometry to non-Euclidean norms.

Abstract

We prove that any self-contracted curve in R 2 endowed with a C 2 and strictly convex norm, has finite length. The proof follows from the study of the curve bisector of two points in R 2 for a general norm together with an adaptation of the argument used in [2].