Measure on surface with geodesic boundary
The measure on pants is a convex function of three variables. We can also extend the definition of this measure for any surface with geodesic boundary. For instance, we can begin with a one-holed torus or four-holed sphere. That would be great if we can show that the measure also convex for any surface. There is a difficulty that we need to confront is that the measure for a general surface does not depend only on the length of geodesics at the boundary, it will also depend on the simple length spectrum of the surface. We can ask if the maximum or minimum of the measure will depend on the systole of the surface. We will try to answer this question by first working on the one-holed torus and four-holed sphere.
We can only define a new measure on pants, for instance, instead of measuring all the vectors whose associated graphs have two vertices, we can measure all the vectors whose associated graphs have n vertices. Then if n tends to infinity, the measure converges to 0. We can prove that for any even number n, the measure is a convex function of three variables. We can also ask that if there is number n so that we can obtain a better upper bound on the growth of the number of pants.