  The standard Wojtkowski-Markarian-Donnay-Bunimovich technique for the
hyperbolicity of focusing or mixed billiards in the plane requires the diameter
of a billiard table to be of the same order as the largest ray of curvature
along the focusing boundary. This is due to the physical principle that is used
in the proofs, the so-called defocusing mechanism of geometrical optics. In
this paper we construct examples of hyperbolic billiards with a focusing
boundary component of arbitrarily small curvature whose diameter is bounded by
a constant independent of that curvature. Our proof employs a nonstardard cone
bundle that does not solely use the familiar dispersing and defocusing
mechanisms.
