Smooth interpolating curves of prescribed length and minimum curvature

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 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 R", n > 2. Introduction. Garrett Birkhoff has conjectured the existence of a curve of minimum integral mean square curvature, interpolating a finite set of planar points, provided a feasible upper bound is prescribed for the length of the interpolating curve. The purpose of this note is to confirm this con- jecture. It is dedicated to Professor Birkhoff on the occasion of his sixty- fourth birthday. That this condition of prescribed length is necessary was shown earlier by Birkhoff and de Boor (l). It is now seen to be a natural necessary and sufficient condition for the existence of smooth interpolating curves of minimum curvature provided the point set is not collinear.