  We focus on the problem of manifold estimation: given a set of observations
sampled close to some unknown submanifold $M$, one wants to recover information
about the geometry of $M$. Minimax estimators which have been proposed so far
all depend crucially on the a priori knowledge of some parameters quantifying
the underlying distribution generating the sample (such as bounds on its
density), whereas those quantities will be unknown in practice. Our
contribution to the matter is twofold: first, we introduce a one-parameter
family of manifold estimators $(\hat{M}_t)_{t\geq 0}$ based on a localized
version of convex hulls, and show that for some choice of $t$, the
corresponding estimator is minimax on the class of models of $C^2$ manifolds
introduced in [Genovese et al., Manifold estimation and singular deconvolution
under Hausdorff loss]. Second, we propose a completely data-driven selection
procedure for the parameter $t$, leading to a minimax adaptive manifold
estimator on this class of models. This selection procedure actually allows us
to recover the Hausdorff distance between the set of observations and $M$, and
can therefore be used as a scale parameter in other settings, such as tangent
space estimation.
