  Closed billiard trajectories in a polygon in the hyperbolic plane can be
coded by the order in which they hit the sides of the polygon. In this paper,
we consider the average length of cyclically related closed billiard
trajectories in ideal hyperbolic polygons and prove the conjecture that this
average length is minimized for regular hyperbolic polygons. The proof uses a
strict convexity property of the geodesic length function in Teichm\"uller
space with respect to the Weil-Petersson metric, a fundamental result
established by Wolpert.
