Smooth Interpolating Curves of Prescribed Length and Minimum Curvature
Joseph W. Jerome · Proceedings of the American Mathematical Society · 1975
It is shown that, among all smooth curves of length not exceeding a prescribed upper bound which interpolate a finite set of planar points, there is at least one which minimizes the curvature in the ${L^2}$ sense. Thus, we show to be sufficient for the solution of the problem of minimum curvature a condition, viz., prescribed length, which has been known to be necessary for at least a decade. The proof extends immediately to curves in ${{\mathbf {R}}^n},n > 2$.