Speed of convergence in first passage percolation and geodesicity of the average distance

We give an elementary proof that Talagrand's sub-Gaussian concentration inequality implies a limit shape theorem for first passage percolation on any Cayley graph of Z^d, with a bound on the speed of convergence that slightly improves Alexander's bounds. Our approach, which does not use the subadditive theorem, is based on proving that the average distance is close to being geodesic. Our key observation, of independent interest, is that the problem of estimating the rate of convergence for the average distance is equivalent (in a precise sense) to estimating its "level of geodesicity".