  We derive non-asymptotic minimax bounds for the Hausdorff estimation of
$d$-dimensional submanifolds $M \subset \mathbb{R}^D$ with (possibly) non-empty
boundary $\partial M$. The model reunites and extends the most prevalent
$\mathcal{C}^2$-type set estimation models: manifolds without boundary, and
full-dimensional domains. We consider both the estimation of the manifold $M$
itself and that of its boundary $\partial M$ if non-empty. Given $n$ samples,
the minimax rates are of order $O\bigl((\log n/n)^{2/d}\bigr)$ if $\partial M =
\emptyset$ and $O\bigl((\log n/n)^{2/(d+1)}\bigr)$ if $\partial M \neq
\emptyset$, up to logarithmic factors. In the process, we develop a
Voronoi-based procedure that allows to identify enough points $O\bigl((\log
n/n)^{2/(d+1)}\bigr)$-close to $\partial M$ for reconstructing it.
