In an open convex body one can define various geometries. One is associated to
convex projective structures on compact manifolds and is known as Hilbert
geometry.
Curvature as we know it in Riemannian geometry is not really available for
this structure, nevertheless these geometries share a lot with the
universal covers of non positive compact manifolds.
In this talk I would like to present my answer to a generalisation of a
famous result, in this latest case, relating the volume entropy with the
Hausdorff dimension of the Gromov Boundary.
This work involves many years and my collaborators among which are Berck,
Bernig, Faifman, Verovic and Walsh.