Relative growth rate and contact Banach-Mazur distance

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.