The existence of caustics for a billiard problem in a convex domain

A system of caustics is found for a plane convex domain with a sufficiently smooth boundary; the caustics are close to the boundary and occupy a set of positive measure. A caustic is a convex smooth curve lying in the domain and possessing the property that a tangent to it becomes another tangent to the same curve after reflection from the boundary according to the law of geometrical optics.