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In this paper, we define non-linear versions of Banach–Mazur distance in the contact geometry set-up, called contact Banach–Mazur distances and denoted by d CBM . Explicitly, we consider the following two set-ups, either on a contact manifold W × S 1 where W is a Liouville manifold, or a closed Liouville-fillable contact manifold M. The inputs of d CBM are different in these two cases. In the former case the inputs are (contact) star-shaped domains of W × S 1 which correspond to the homotopy classes of positive contact isotopies, and in the latter case the inputs are contact 1-forms of M inducing the same contact structure. In particular, the contact Banach–Mazur distance d CBM defined in the former case is motivated by the concept, relative growth rate, which was originally defined and studied by Eliashberg–Polterovich. The main results are the large-scale geometric properties in terms of d CBM . In addition, we propose a quantitative comparison between elements in a certain subcategory of the derived categories of sheaves of modules (over certain topological spaces). This is based on several important properties of the singular support of sheaves and Guillermou–Kashiwara–Schapira's sheaf quantization. [ABSTRACT FROM AUTHOR]