Preprints and Recent Publications

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Three papers on Hilbert geometries

The year 2009 was a good year for Hilbert geometries. It has been finally settled (independently) by A. Bernig and C. Vernicos that

The Hilbert geometry in the interior of an n-dimensional convex body is bilipschitz equivalent to an n-dimensional normed space if and only if the body is a polytope.

In another interesting paper, B. Lemmens and C. Walsh have shown that if the subgroup of projective transformations that sends a polytope to itself is properly contained in the isometry group of its Hilbert geometry, then the polytope must be an n-simplex (n > 1). Moreover, they show that the subgroup of projective transformations that sends an n-simple (n > 1) to itself is an index-two subgroup of the isometry group of its Hilbert geometry.

Three papers on systolic inequalities

Roughly speaking, a systolic inequality is a lower bound for the volume of a closed Riemannian or Finsler manifold in terms of the length of a suitable closed geodesic. There is a large body of work on systolic inequalities on Riemannian manifolds (Loewner, Pu, Berger, Gromov, Katz, Babenko, Bangert, ...), but the Finsler theory has been slow to take off. I think this is mostly because questions about systolic rigidity or freedom (i.e., whether one can expect non-trivial systolic inequalities at all) for Riemannian manifolds immediately apply to reversible Finsler metrics, and because questions about sharp inequalities already have the reputation of being hopeless in the Riemannian case. Nevertheless, the beginnings of a Finsler theory are being set. Ivanov and Sabourau have, respectively, proved sharp systolic inequalities for reversible Finsler metrics on the projective plane and the two-dimensional torus:

The Holmes-Thompson volume of a reversible Finsler metric on the two-torus or the projective plane is at least equal to $2/pi$ times the square of the length of the shortest non-contractible closed geodesic.

In the non-reversible case the only reference that seems to be available is a recent paper by Alvarez Paiva and Balacheff where it is shown that (not-necessarily reversible) Finsler manifolds all of whose geodesics are closed and of the same length satisfy an infinitesimal isosystolic inequality to all orders. The final section of this paper contains a short survey of what is known in Finsler systolic geometry and explains the importance of the choice of volume for non-reversible metrics.