  Given a sample $Y$ from an unknown manifold $X$ embedded in Euclidean space,
it is possible to recover the homology groups of $X$ by building a
Vietoris--Rips or \v{C}ech simplicial complex on top of the vertex set $Y$.
However, these simplicial complexes need not inherit the metric structure of
the manifold, in particular when $Y$ is infinite. Indeed, a simplicial complex
is not even metrizable if it is not locally finite. We instead consider metric
thickenings, called the \emph{Vietoris--Rips} and \emph{\v{C}ech thickenings},
which are equipped with the 1-Wasserstein metric in place of the simplicial
complex topology. We show that for Euclidean subsets $X$ with positive reach,
the thickenings satisfy metric analogues of Hausmann's theorem and the nerve
lemma (the metric Vietoris--Rips and \v{C}ech thickenings of $X$ are homotopy
equivalent to $X$ for scale parameters less than the reach). To our knowledge
this is the first version of Hausmann's theorem for Euclidean submanifolds (as
opposed to Riemannian manifolds), and our result also extends to non-manifold
shapes (as not all sets of positive reach are manifolds). In contrast to
Hausmann's original proof, our homotopy equivalence is a deformation
retraction, is realized by canonical maps in both directions, and furthermore
can be proven to be a homotopy equivalence via simple linear homotopies from
the map compositions to the corresponding identity maps.
