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

*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.*
* 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**

*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.*
* (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.