  We investigate the terms arising in an identity for hyperbolic surfaces
proved by Luo and Tan, namely showing that they vary monotonically in terms of
lengths and that they verify certain convexity properties. Using these
properties, we deduce two results. As a first application, we show how to
deduce a theorem of Thurston which states, in particular for closed hyperbolic
surfaces, that if a simple length spectrum "dominates" another, then in fact
the two surfaces are isometric. As a second application, we show how to find
upper bounds on the number of pairs of pants of bounded length that only depend
on the boundary length and the topology of the surface.
