Plane R-curves II

ABSTRACT

Let ${\mathrm{\Gamma }}_R$ be the class of plane, oriented, rectifiable curves γ, such that, for almost every $x\in \gamma$, the part of γ preceding x is outside the open disk of radius R, centered in $x+R{\mathbf{t}}_x$, where ${\mathbf{t}}_x$ is the unit tangent vector at x. In Longinetti et al. [Plane R-curves and their steepest descent properties I. Appl Anal. DOI: 10.1080/00036811.2018.1466278], the present authors have obtained bounds for the length and the detour for $C^1$ regular curves in ${\mathrm{\Gamma }}_R$. These bounds are proved here for all curves in ${\mathrm{\Gamma }}_R$.

COMMUNICATED BY:

• Donal O'Regan

Keywords:

• Steepest descent curves
• sets with positive reach
• length of curves
• detour

AMS Subject Classifications:

• Primary: 28A75
• 52A30
• Secondary: 34A26
• 34A60

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work has been partially supported by GNAMPA of INDAM, Istituto Nazionale di Alta Matematica.