Principal normal indicatrix of a closed space curves.

Considering the differential geometry of (closed) space curves it is usually assumed that curvature never vanishes. A natural extension of the Frenet Apparatus, that allows curvature to take both signs as long a curvature and torsion do not vanish simultaneously, is eg. used by W. Fenchel in 1950. At that time it was known that the principal normal indicatrix of a closed space curve fulfilling the Fenchel condition has integrated geodesic curvature equal to an integral multiple of 2*Pi. In 1950 W. Fenchel gave a reformulation of the inverse problem in terms of the convex hulls of a family of closed curves on the unit 2-sphere.
By a more resent theorem due to J. Weiner (1991), which also is proven curve-theoreticly by B. Solomon (1996) it follows that the principal normal indicatrix of a closed space curve with non-vanishing curvature has integrated geodesic curvature zero and contains no sub arc with integrated geodesic curvature Pi.
In this project it is proven that, when restricted to closed curves on the unit 2-sphere with integrated geodesic curvature zero that contains no sub arc with integrated geodesic curvature Pi, then the inverse problem formulated by W. Fenchel always has solutions, if one allows zero and negative curvature of space curves. Furthermore it is explained why this is not true if non-vanishing curvature is required.
By giving examples of closed curves on the unit 2-sphere with integrated geodesic curvature 2*Pi*z, for any integer z, which are not a principal indicatrix of a closed space curve (even when allowing zero and negative curvature) the general answer to W. Fenchels question is found to be to the negative.