On the total curvature of curves in a Minkowski space

We consider simple closed curves in a Minkowski space. We give bounds of the total Minkowski curvature of the curve in terms of the total Euclidean curvature and of normal curvatures on the indicatrix (supposed to be a central symmetric hypersurface) of the Minkowski norm. Corollaries of this result provide analogues to Fenchel and Fary-Milnor theorems. We also give an upper bound of the Minkowski length of a simple closed curve contained in a Minkowski ball of radius R, in terms of the total Minkowski curvature and of normal curvatures on the indicatrix. Whenever the Minkowski space is Euclidean our results reduce to the classical ones.