Plane R-curves and their steepest descent properties I

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