  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.
