The main purpose of this dissertation is to understand the large scale geometry of the space of contactforms supporting a given overtwisted contact structure $\xi$ on a contact manifold $Y$. In order to achieve this, inspired by analogous work of Ostrover and Polterovich which was further developed by Usher, Gutt, Zhang and Stojisavljevic, we define a distance on the space of contact forms $\alpha$ with $ker(\alpha) = \xi$ which we denote by $C^Y_\xi$. This distance is called the contact Banach-Mazur distance. It can be viewed as the analogue of of the symplectic Banach-Mazur distance for Liouville domains, in the case of overtwisted or more generally non-fillable contact manifolds. Viewing contact homology algebra as a persistence module, focusing purely on the overtwisted case and exploiting the fact that the contact homology of overtwisted contact structures vanishes, allows us to bi-Lipschitz embed part of the 2-dimensional Euclidean space into the space of overtwisted contact forms $C^Y_\xi$ supporting a given contact structure on a smooth closed manifold $Y$ . This is the main result of this work.