The boundness of distance between two sets of fixed volume inside the multidimensional ball or cube

Abstract

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