  Given a Lagrangian submanifold in linear symplectic space, its tangent sweep
is the union of its (affine) tangent spaces, and its tangent cluster is the
result of parallel translating these spaces so that the foot point of each
tangent space becomes the origin. This defines a multivalued map from the
tangent sweep to the tangent cluster, and we show that this map is a local
symplectomorphism (a well known fact, in dimension two).
  We define and study the outer billiard correspondence associated with a
Lagrangian submanifold. Two points are in this correspondence if they belong to
the same tangent space and are symmetric with respect to its foot pointe. We
show that this outer billiard correspondence is symplectic and establish the
existence of its periodic orbits. This generalizes the well studied outer
billiard map in dimension two.

Symplectic Geometry