The Isoperimetric Problem of the Convex Hull of a Closed Space Curve

I. Let C be a closed rectifiable curve or an open rectifiable arc in the 77-dimensional Euclidean space En. Let C, V(C) and L(C) denote respectively the closed convex hull of C, the volume of C and the length of C. Let (c, n), or (0, n), stand for the problem of maximizing V(C), subject to the condition L(C)=const, for a closed curve C, or an open arc C. It is clear that the solution is to be given only up to a similarity and a rigid motion, possibly followed by a reflexion. On closer examination it turns out that there are four classes of problems: (c, 2n), (0, 2t7), (0, 2n + l) and (c, 2n + l). Under certain restrictive assumptions (c, 2n) has been solved by Schoenberg [l]. Generalizing the Fourier-series isoperimetric method of Hurwitz [2], he proves that the solution C of (c, 2n) is given parametrically by