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Applicable Analysis
An International Journal
Volume 98, 2019 - Issue 14
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Articles

Plane R-curves and their steepest descent properties I

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Pages 2497-2510 | Received 07 Jun 2017, Accepted 14 Apr 2018, Published online: 29 Apr 2018
 

ABSTRACT

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

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Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

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

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