Let $S_g$ be the closed surface of genus $g$ and $T_g$ its Teichmueller space. Given a closed curve $\gamma$ in $S_g$ we consider the so-called length function; it is namely the function which associates to a point $X\in T_g$ in Teichmueller space the length of the unique geodesic in $X$ freely homotopic to $\gamma$. Kerckhoff and Wolpert proved that the geodesic length functions are convex along earthquake paths and Weil-Peterson geodesics respectively. This convexity was the key-point in the solution of the Nielsen realization problem.
The goal of this talk is to prove that the length functions are also convex with respect to (well-chosen) Fenchel-Nielsen coordinates. It is a pleasent fact that the proof is purely synthetic.
This is joint work with Mladen Bestvina and Ken Bromberg.