Growth estimates on optimal transport maps via concentration inequalities

We give an alternative proof and some extensions of results of Carlier, Figalli and Santambrogio on polynomial upper bounds on the Brenier map between probability measures under various conditions on the densities. The proofs are based on the monotonicity of the map and various concentration inequalities, as already used by Colombo and Fathi to prove quadratic growth for transport maps from the standard Gaussian onto log-concave measures.