Characterization of Tantrix Curves

Let S1 be the unit circle realized as R/Z. Then a regular closed curve in Rn is a smooth mapping c : S1 → Rn so that the velocity vector c′(t) never vanishes. The tantrix of such a curve is the map t : S1 → Sn−1 given by t(t) = c′(t)/‖c′(t)‖, that is the unit tangent to c parallel translated to the origin. Our goal is to understand which curves t : S1 → Sn−1 can be realized as the tantrix to a regular closed curve. A first step is the following, due to Löwner, is a necessary condition for a curve to be a tantrix. This is attributed to Löwner by both Fenchel [1, p. 39] and Pólya and Szegő [2, Band II S. 165 und 391 Aufgabe, 13.]. I have not been able to track down the original paper of Löwner. Theorem 1.1 (Löwner). If a curve t : S1 → Sn−1 is a tantrix then the origin is in the convex hull of the image of t. This implies that every totally geodesic Sn−2 in Sn−1 meets t in at least one point.