ON CURVES OF MINIMAL LENGTH WITH A CONSTRAINT ON AVERAGE CURVATURE, AND WITH PRESCRIBED INITIAL AND TERMINAL POSITIONS AND TANGENTS

摘要

The purpose of this paper is to prove Theorem 1, which implies that for n 2, an R-geodesic is necessarily a continuously differentiable curve which consists of not more than four pieces, each of which is either a straight line segment or an arc of a circle of radius R. Furthermore, the corollary to Theorem 1 implies that four is the least integer for which this is true. The nature of R-geodesics for n > 3 is open