The total absolute curvature of closed curves with singularities

In this paper, we give a generalization of Fenchel's theorem for closed curves as frontals in Euclidean space Rⁿ. We prove that, for a non-co-orientable closed frontal in Rⁿ, its total absolute curvature is greater than or equal to. It is equal to if and only if the curve is a planar locally L-convex closed frontal whose rotation index is 1/2 or -1/2. Furthermore, if the equality holds and if every singular point is a cusp, then the number N of cusps is an odd integer greater than or equal to 3, and N=3 holds if and only if the curve is simple.