  In this paper, we investigate the growth error bound condition. By using the
proximal point algorithm, we first provide a more accessible and elementary
proof of the fact that Kurdyka-{\L}ojasiewicz conditions imply growth error
bound conditions for convex functions which has been established before via a
subgradient flow. We then extend the result for nonconvex functions.
Furthermore we show that every definable function in an o-minimal structure
must satisfy a growth error bound condition. Finally, as an application, we
consider the heavy ball method for solving convex optimization problems and
propose an adaptive strategy for selecting the momentum coefficient. Under
growth error bound conditions, we derive convergence rates of the proposed
method. A numerical experiment is conducted to demonstrate its acceleration
effect over the gradient method.
