On the rate of convergence of empirical measures in   $\infty$-transportation distance
  We consider random i.i.d. samples of absolutely continuous measures on bounded connected domains. We prove an upper bound on the $\infty$-transportation distance between the measure and the empirical measure of the sample. The bound is optimal in terms of scaling with the number of sample points. 
Nicol\'as Garc\'ia Trillos and Dejan Slep\v{c}ev
math.PR math.FA