  The inverse optimal transport problem is to find the underlying cost function
from the knowledge of optimal transport plans. While this amounts to solving a
linear inverse problem, in this work we will be concerned with the nonlinear
inverse problem to identify the cost function when only a set of marginals and
its corresponding optimal values are given. We focus on absolutely continuous
probability distributions with respect to the $d$-dimensional Lebesgue measure
and classes of concave and convex cost functions. Our main result implies that
the cost function is uniquely determined from the union of the ranges of the
gradients of the optimal potentials. Since, in general, the optimal potentials
may not be observed, we derive sufficient conditions for their identifiability
- if an open set of marginals is observed, the optimal potentials are then
identified via the value of the optimal costs. We conclude with a more in-depth
study of this problem in the univariate case, where an explicit representation
of the transport plan is available. Here, we link the notion of identifiability
of the cost function with that of statistical completeness.
