Coisotropic Displacement and Small Subsets of a Symplectic Manifold

We prove a coisotropic intersection result and deduce the following: Lower bounds on the displacement energy of a subset of a symplectic manifold, in particular a sharp stable energy-Gromov-width inequality. A stable non-squeezing result for neighborhoods of products of unit spheres. Existence of a ``badly squeezable'' set in $R^{2n}$ of Hausdorff dimension at most $d$, for every $n\geq2$ and $d\geq n$. Existence of a stably exotic symplectic form on $R^{2n}$, for every $n\geq2$. Non-triviality of a new capacity, which is based on the minimal symplectic area of a regular coisotropic submanifold of dimension $d$