Plane R-curves and their steepest descent properties I

ABSTRACT

Let ${\mathrm{\Gamma }}_R$ be the class of plane, oriented, rectifiable curves $\gamma$, such that for almost every $x\in \gamma$, the part of $\gamma$ preceding x is outside of the open circle of radius R, centered in $x+R{\mathbf{t}}_x$, where ${\mathbf{t}}_x$ is the unit tangent vector at x. Geometrical properties of the curves $\gamma \in {\mathrm{\Gamma }}_R$ are proved; it is shown also that the length of a regular curve $\gamma \in {\mathrm{\Gamma }}_R$ is bounded by a constant depending upon R and the diameter of $\gamma$ only. The curves $\gamma \in {\mathrm{\Gamma }}_R$ turn out to be steepest descent curves for real-valued functions with sublevel sets of reach greater than R.

KEYWORDS:

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

AMS SUBJECT CLASSIFICATIONS:

• Primary: 49Q15
• 52A30
• Secondary: 34A26
• 34A60

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work has been partially supported by INDAM-GNAMPA(2016).