On the strict Arnold chord property and coisotropic submanifolds of complex projective space

Let be a contact form on a manifoldM, and L M a closed Legendrian submanifold. I prove that L intersects some characteristic for at least twice if all characteristics are closed and of the same period, and embeds nicely into the product of R2n and an exact symplectic manifold. As an application of the method of proof, the minimal action of a regular closed coisotropic submanifold of complex projective space is at most /2. This yields an obstruction to presymplectic embeddings, and in particular to Lagrangian embeddings.