On the Strict Arnold Chord Property and Coisotropic Submanifolds of Complex Projective Space

Let ${\alpha }$ be a contact form on a manifold $M$, and $L\subseteq M$ a closed Legendrian submanifold. I prove that $L$ intersects some characteristic for ${\alpha }$ at least twice if all characteristics are closed and of the same period, and ${\alpha }$ embeds nicely into the product of ${\BBR}^{2n}$ 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 $\pi /2$. This yields an obstruction to presymplectic embeddings, and in particular to Lagrangian embeddings.