The Fenchel-type inequality in the 3-dimensional Lorentz space and a Crofton formula

We generalize the Fenchel theorem to strong spacelike (which means that the tangent vector and the curvature vector span a spacelike 2-plane at each point) closed curves with index 1 in the 3-dimensional Lorentz space, showing that the total curvatures must be less than or equal to . A similar generalization of the Fary-Milnor theorem is also obtained. We establish the Crofton formula on the de Sitter 2-sphere which implies the above results.SCI(E)ARTICLEyen@pku.edu.cn; maxiang@math.pku.edu.cn; wdh_nick_123456@126.com3249-2595