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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   $\mathbb {R}^{2n}$   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.

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