Resumen de A note on Finsler-Minkowski norms
By a Finsler--Minkowski norm, we mean a function on a real vector space which is positively homogeneous and positive on the non-zero vectors. We suppose that its metric tensor, i.e., the second derivative of its square is non-degenerate. Then we show that it is automatically positive definite.
The main idea of the proof is as follows. We suppose, without loss of generality, that our vector space is a Euclidean n-space. The unit sphere is a compact hypersurface, and therefore it has a point, the furthermost one from the origin, in which all the n-1 principal curvatures have the same sign. Finally, we establish a relation between the signs of these principal curvatures and the signature of the metric tensor.
