  Let $\mu$ be a probability measure on $\mathbb{R}^d$ and $\mu_N$ its
empirical measure with sample size $N$. We prove a concentration inequality for
the optimal transport cost between $\mu$ and $\mu_N$ for radial cost functions
with polynomial local growth, that can have superpolynomial global growth. This
result generalizes and improves upon estimates of Fournier and Guillin. The
proof combines ideas from empirical process theory with known concentration
rates for compactly supported $\mu$. By partitioning $\mathbb{R}^d$ into
annuli, we infer a global estimate from local estimates on the annuli and
conclude that the global estimate can be expressed as a sum of the local
estimate and a mean-deviation probability for which efficient bounds are known.
