  The empirical optimal transport (OT) cost between two probability measures
from random data is a fundamental quantity in transport based data analysis. In
this work, we derive novel guarantees for its convergence rate when the
involved measures are different, possibly supported on different spaces. Our
central observation is that the statistical performance of the empirical OT
cost is determined by the less complex measure, a phenomenon we refer to as
lower complexity adaptation of empirical OT. For instance, under Lipschitz
ground costs, we find that the empirical OT cost based on $n$ observations
converges at least with rate $n^{-1/d}$ to the population quantity if one of
the two measures is concentrated on a $d$-dimensional manifold, while the other
can be arbitrary. For semi-concave ground costs, we show that the upper bound
for the rate improves to $n^{-2/d}$. Similarly, our theory establishes the
general convergence rate $n^{-1/2}$ for semi-discrete OT. All of these results
are valid in the two-sample case as well, meaning that the convergence rate is
still governed by the simpler of the two measures. On a conceptual level, our
findings therefore suggest that the curse of dimensionality only affects the
estimation of the OT cost when both measures exhibit a high intrinsic
dimension. Our proofs are based on the dual formulation of OT as a maximization
over a suitable function class $\mathcal{F}_c$ and the observation that the
$c$-transform of $\mathcal{F}_c$ under bounded costs has the same uniform
metric entropy as $\mathcal{F}_c$ itself.
