In this talk, I will give a survey in convex geometries associated with Teichmuller space. In particular, an emphasis is put on the Weil-Petersson geometry, where deformations of hyperbolic metrics on a given topological surface is measured by the $L^2$ norm of the deformation tensors. Also mentioned is the several convex geometries naturally occurring in the investigation of Teichmuller space, including the geometries of the Teichmuller metric and the Thurston metric.