  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".
