Displacement convexity for point processes and an application

I will explain how coisotropic submanifolds of symplectic manifolds can be distinguished among all submanifolds by a criterion ("local rigidity") related to the Hofer energy necessary to disjoin open sets from them. This criterion is invariant under symplectic homeomorphisms, leading to a simplified proof of the Humiliere-Leclercq-Seyfaddini theorem that a symplectic-homeomorphic image of a coisotropic submanifold that is smooth is coisotropic. Moreover much of this picture extends to the contact context, allowing one to extend the class of C^0-limits of contactomorphisms which are known to map Legendrian submanifolds to Legendrian submanifolds.