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Coisotropic Displacement and Small Subsets of a Symplectic Manifold

We prove a coisotropic intersection result and deduce the following: 1. Lower bounds on the displacement energy of a subset of a symplectic manifold, in particular a sharp stable energy-Gromov-width inequality. 2. A stable non-squeezing result for neighborhoods of products of unit spheres. 3. Existence of a "badly squeezable" set in $\mathbb{R}^{2n}$ of Hausdorff dimension at most $d$, for every $n\geq2$ and $d\geq n$. 4. Existence of a stably exotic symplectic form on $\mathbb{R}^{2n}$, for every $n\geq2$. 5. Non-triviality of a new capacity, which is based on the minimal symplectic area of a regular coisotropic submanifold of dimension $d$.