The size of an overtwisted disk
We introduce a notion of size for an overtwisted disk, defined in terms of the Shelukhin-Hofer energy required to stabilise Legendrians in its vicinity. We compute this size for the standard $3$-dimensional overtwisted disk, which we use to prove that certain explicit product hyperdisks in higher-dimensional contact manifolds are overtwisted (and their overtwisted size can be computed explicitly as well). We then use the overtwisted size to establish a number of quantitative results on contact flexibility. Concretely, we (1) obtain quantitative $h$--principles for Legendrians in the complement of an overtwisted disk, (2) construct overtwisted-at-infinity contact manifolds whose Legendrian Shelukhin-Hofer metric vanishes, (3) estimate overtwisted size under contact products and under products with Liouville manifolds, and (4) derive action bounds for contractible Reeb orbits in closed overtwisted contact manifolds.