A note on Finsler-Minkowski norms

• Autores: Rezso L. Lovas
• Localización: Houston journal of mathematics, ISSN 0362-1588, Vol. 33, Nº 3, 2007, págs. 701-707
• Idioma: inglés
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• Resumen

  • 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.