As a deformation space of hyperbolic metrics defined on a closed surface of genus , Teichmüller space can be regarded as a convex set with respect to the Weil--Petersson geometry. The convexity is used to construct a new distance function, called the Weil–Petersson Funk metric, which is a weak metric, lacking the symmetry and non-degeneracy conditions. The distance between two points is defined as a supremum of functions on the pair of points indexed by the set of the simple closed curves of the given surface. This set acts as the index set of the supporting hyperplanes of Teichmüller space regarded as a Weil–Petersson convex body.
This Funk-type construction also appears in defining the two well-known distance functions on Teichmüller space: the Teichmüller metric and the Thurston metric. We emphasize in this chapter that the underlying idea for the three distance functions is to treat the Teichmüller space as a convex body.