  In the symplectic realm, a distance between open starshaped domains in
Liouville manifolds was recently defined. This is the symplectic Banach-Mazur
distance. It was proposed by Ostrover and Polterovich and developed by
Ostrover, Polterovich, Usher, Gutt, Zhang and Stojisavljevi\'c. The natural
question is, can an analogous distance in the contact realm be defined? One
idea is to define the distance on contact hypersurfaces of Liouville manifolds
and another one on contact forms supporting isomorphic contact structures.
Rosen and Zhang recently defined such a distance working with manifolds that
are prequantizations of Liouville manifolds. They also considered a distance on
contact forms supporting the same contact structure on a contact manifold $Y$.
This allowed them to view the space of contact forms supporting isomorphic
contact structures on a manifold $Y$ as a pseudometric space, study its
properties, and derive interesting results. In this work, we do something
similar, yet the distance we define is less restrictive. Moreover, 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 supporting a given contact structure on a smooth closed manifold $Y$.
