  We show that for any compact convex set $K$ in $\mathbb{R}^d$ and any finite
family $\mathcal{F}$ of convex sets in $\mathbb{R}^d$, if the intersection of
every sufficiently small subfamily of $\mathcal{F}$ contains an isometric copy
of $K$ of volume $1$, then the intersection of the whole family contains an
isometric copy of $K$ scaled by a factor of $(1-\varepsilon)$, where
$\varepsilon$ is positive and fixed in advance. Unless $K$ is very similar to a
disk, the shrinking factor is unavoidable. We prove similar results for affine
copies of $K$. We show how our results imply the existence of randomized
algorithms that approximate the largest copy of $K$ that fits inside a given
polytope $P$ whose expected runtime is linear on the number of facets of $P$.
