  A metric thickening of a given metric space $X$ is any metric space admitting
an isometric embedding of $X$. Thickenings have found use in applications of
topology to data analysis, where one may approximate the shape of a dataset via
the persistent homology of an increasing sequence of spaces. We introduce two
new families of metric thickenings, the $p$-Vietoris-Rips and $p$-\v{C}ech
metric thickenings for all $1\le p\le \infty$, which include all measures on
$X$ whose $p$-diameter or $p$-radius is bounded from above, equipped with an
optimal transport metric. The $p$-diameter (resp. $p$-radius) of a measure is a
certain $\ell_p$ relaxation of the usual notion of diameter (resp. radius) of a
subset of a metric space. These families recover the previously studied
Vietoris-Rips and \v{C}ech metric thickenings when $p=\infty$. As our main
contribution, we prove a stability theorem for the persistent homology of
$p$-Vietoris-Rips and $p$-\v{C}ech metric thickenings, which is novel even in
the case $p=\infty$. In the specific case $p=2$, we prove a Hausmann-type
theorem for thickenings of manifolds, and we derive the complete list of
homotopy types of the $2$-Vietoris-Rips thickenings of the $n$-sphere as the
scale increases.
