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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.

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    $\begingroup$ There are in fact at least two different possible "canonical" volume forms. See library.msri.org/books/Book50/files/02AT.pdf $\endgroup$
    – Deane Yang
    Aug 16, 2017 at 14:56
  • $\begingroup$ @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? $\endgroup$
    – Ali Taghavi
    Aug 16, 2017 at 17:14
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    $\begingroup$ @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. $\endgroup$
    – alvarezpaiva
    Aug 18, 2017 at 20:15
  • $\begingroup$ @alvarezpaiva Thanks for this very interesting concept "John ellipsoid". BTW is there a norm on $\mathbb{R}^2$ whose isometry group is the whole $\pm sl(2,\mathbb{R})$? $\endgroup$
    – Ali Taghavi
    Aug 19, 2017 at 9:47
  • $\begingroup$ @AliTaghavi, no, same reasoning: isometries of a finite-dimensional normed space are isometries of an adapted Euclidean metric. They are all Euclidean transformation. $\endgroup$
    – alvarezpaiva
    Aug 19, 2017 at 12:50

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In fact there are very many ways to provide a Finsler manifold with a "canonical" volume. Personally I've gone from thinking that this is a nuisance and trying to pin down which one is really the best to thinking that this is part of the landscape and should be accepted.

There is a very good notion of volume that goes by the name of "Holmes-Thompson" volume, but it was also introduced by Dazur and, in my opinion, half-heartedly studied by Busemann. You can find almost all that is known about it in the book Minkowski Geometry by Thompson and in the paper Deane Yang linked to in his comment.

In the paper by Busemann that you mention, he claims the Hausdorff measure of the Finsler manifold, viewed as a metric space, is the right notion of volume. It is a very interesting notion, of course, but it has its quirks : totally geodesic submanifolds are not minimal, integral geometry goes out the window, some volume filling results fail, etc. It is also not great for non-reversible Finsler metrics. The Holmes-Thompson is nicer in this respect too because it is more sensitive to non-reversibility.

Although it is too long to explain here, my viewpoint has changed and I think that one can and should consider, always with some measure and with a lot of good taste, all the natural notions of volume. Sometimes keeping the same questions and changing the notion of volume opens up new vistas and allows you to tie Finsler geometry to other fields, which is what I think is needed most. Check for example this paper. Behind the paper is the idea that there is a different geometry of numbers for every natural notion of volume. The classical results are for Hausdorff measure and this paper makes the case that for the Holmes-Thompson measure you also get an interesting theory (which was, by the way, forseen by Mahler).

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