Estimate of the distance between two bodies inside an $n$-dimensional unit cube and a ball

The problem of estimation of the distance between two bodies of volume $\varepsilon$ located inside an $n$-dimensional body $B$ of unit volume where $n \to \infty$ is considered. In some cases such distances are bounded by a function of $\varepsilon$ not dependent on $n$. The cases when $B$ is a sphere or a cube are considered.