  We consider the problem of optimal incomplete transportation between the
empirical measure on an i.i.d. uniform sample on the d-dimensional unit cube
$[0,1]^d$ and the true measure. This is a family of problems lying in between
classical optimal transportation and nearest neighbor problems. We show that
the empirical cost of optimal incomplete transportation vanishes at rate
$O_P(n^{-1/d})$, where n denotes the sample size. In dimension $d\geq3$ the
rate is the same as in classical optimal transportation, but in low dimension
it is (much) higher than the classical rate.
