  We study convex and quasiconvex functions on a metric graph. Given a set of
points in the metric graph, we consider the largest convex function below the
prescribed datum. We characterize this largest convex function as the unique
largest viscosity subsolution to a simple differential equation, $u''=0$ on the
edges, plus nonlinear transmission conditions at the vertices. We also study
the analogous problem for quasiconvex functions and obtain a characterization
of the largest quasiconvex function that is below a given datum.
