# Volume form induced by a Finsler metric

I'm interested in knowing more about the volume form canonically induced by a Finsler metric.

I've found some reasoning about it in this article http://www.ams.org/journals/bull/1950-56-01/S0002-9904-1950-09332-X/home.html but I was wondering if someone could point out a more recent source with results explained in a clearer way.

• There are in fact at least two different possible "canonical" volume forms. See library.msri.org/books/Book50/files/02AT.pdf – Deane Yang Aug 16 '17 at 14:56
• @User28341 is there a norm on $\mathbb{R}^2$ which does not satisfy the parallelogram equality but every isometry of this norm, preserves the standard volum form? – Ali Taghavi Aug 16 '17 at 17:14
• @AliTaghavi? all norms satisfy this. Every isometry for a norm in $\mathcal{R}^n$ is an isometry for the Euclidean metric associated to its John (or Legendre, or Binet) ellipsoid and therefore preserves the standard volume form. – alvarezpaiva Aug 18 '17 at 20:15
• @alvarezpaiva Thanks for this very interesting concept "John ellipsoid". BTW is there a norm on $\mathbb{R}^2$ whose isometry group is the whole $\pm sl(2,\mathbb{R})$? – Ali Taghavi Aug 19 '17 at 9:47
• @AliTaghavi, no, same reasoning: isometries of a finite-dimensional normed space are isometries of an adapted Euclidean metric. They are all Euclidean transformation. – alvarezpaiva Aug 19 '17 at 12:50