  In this paper, we define a non-linear version of Banach-Mazur distance in the
contact geometry set-up, called contact Banach-Mazur distance and denoted by
$d_{\rm CBM}$. Explicitly, we consider the following two set-ups, either on a
contact manifold $W \times S^1$ where $W$ is a Liouville manifold, or a closed
Liouville-fillable contact manifold $M$. The inputs of $d_{\rm CBM}$ are
different in these two cases. In the former case the inputs are (contact)
star-shaped domains of $W \times S^1$, and in the latter case the inputs are
contact 1-forms of $M$. In particular, the contact Banach-Mazur distance
$d_{\rm CBM}$ defined in the former case is motivated by the concept, relative
growth rate, which was originally defined and studied by Eliashberg and
Polterovich. In addition, we investigate the relations of $d_{\rm CBM}$ to
various numerical measurements in contact geometry and symplectic geometry, for
instance, contact shape invariant, (coarse) symplectic Banach-Mazur distance.
Moreover, we obtain several large-scale geometric properties in terms of
$d_{\rm CBM}$. Finally, we propose a quantitative comparison between elements
in the derived categories of sheaves of modules (over certain topological
spaces). This is based on several important properties of the singular support
of sheaves.

Symplectic Geometry;Metric Geometry