  We use persistence modules and their corresponding barcodes to quantitatively
distinguish between different fiberwise star-shaped domains in the cotangent
bundle of a fixed manifold. The distance between two fiberwise star-shaped
domains is measured by a non-linear version of the classical Banach-Mazur
distance, called symplectic Banach-Mazur distance and denoted by $d_{SBM}.$ The
relevant persistence modules come from filtered symplectic homology and are
stable with respect to $d_{SBM}.$ Our main focus is on the space of unit codisc
bundles of orientable surfaces of positive genus, equipped with Riemannian
metrics. We consider some questions about large scale-geometry of this space
and in particular we give a construction of a quasi-isometric embedding of
$(\mathbb{R}^n,|\cdot |_\infty)$ into this space for all $n\in \mathbb{N}.$ On
the other hand, in the case of domains in $T^*S^2$, we can show that the
corresponding metric space has infinite diameter. Finally, we discuss the
existence of closed geodesics whose energies can be controlled.
