  Given a sample of points $X$ in a metric space $M$ and a scale $r>0$, the
Vietoris-Rips simplicial complex $\mathrm{VR}(X;r)$ is a standard construction
to attempt to recover $M$ from $X$ up to homotopy type. A deficiency of this
approach is that $\mathrm{VR}(X;r)$ is not metrizable if it is not locally
finite, and thus does not recover metric information about $M$. We attempt to
remedy this shortcoming by defining a metric space thickening of $X$, which we
call the \emph{Vietoris-Rips thickening} $\mathrm{VR}^m(X;r)$, via the theory
of optimal transport. When $M$ is a complete Riemannian manifold, or
alternatively a compact Hadamard space, we show that the the Vietoris-Rips
thickening satisfies Hausmann's theorem ($\mathrm{VR}^m(M;r)\simeq M$ for $r$
sufficiently small) with a simpler proof: homotopy equivalence
$\mathrm{VR}^m(M;r)\to M$ is canonically defined as a center of mass map, and
its homotopy inverse is the (now continuous) inclusion map
$M\hookrightarrow\mathrm{VR}^m(M;r)$. Furthermore, we describe the homotopy
type of the Vietoris-Rips thickening of the $n$-sphere at the first positive
scale parameter $r$ where the homotopy type changes.

Metric Geometry;Algebraic Topology;Geometric Topology