We investigate the possibility of approximating a function on a compact set K of the complex plane in such a way that the rate of approximation is almost optimal on K, and the rate inside the interior of K is faster than on the whole of K. We show that if K has an external angle smaller than π at some point z_o∈δK, then geometric convergence inside K is possible only for functions that are analytic at z_o. We also consider the possibility of approximation rates of the form exp(-cn^β), for approximation inside K, where β is related to the largest external angle of K. It is also shown that no matter how slowly the sequence {γ_n} tends to zero, there is a K and a Lip β, β<1, function f such that approximation inside K cannot have order {γ_n}.