  Assume that we observe i.i.d.~points lying close to some unknown
$d$-dimensional $\mathcal{C}^k$ submanifold $M$ in a possibly high-dimensional
space. We study the problem of reconstructing the probability distribution
generating the sample. After remarking that this problem is degenerate for a
large class of standard losses ($L_p$, Hellinger, total variation, etc.), we
focus on the Wasserstein loss, for which we build an estimator, based on kernel
density estimation, whose rate of convergence depends on $d$ and the regularity
$s\leq k-1$ of the underlying density, but not on the ambient dimension. In
particular, we show that the estimator is minimax and matches previous rates in
the literature in the case where the manifold $M$ is a $d$-dimensional cube.
The related problem of the estimation of the volume measure of $M$ for the
Wasserstein loss is also considered, for which a minimax estimator is
exhibited.
