A Symplectically Non-Squeezable Small Set, and the Regular Coisotropic Capacity

Abstract

We prove that for n ≥ 2 there exists a compact subset X of the closed ball in R2n of radius √2, such that X has Hausdorff dimension n and does not symplectically embed into the standard open symplectic cylinder. The second main result is a lower bound on the d-th regular coisotropic capacity, which is sharp up to a factor of 3. For an open subset of a geometrically bounded, as- pherical symplectic manifold, this capacity is a lower bound on its displacement energy. The proofs of the results involve a certain Lagrangian submanifold of linear space, which was considered by M. Audin and L. Polterovich.