Dimension-free estimates on distances between subsets of volume $\epsilon$ inside a unit-volume body

Average distance between two points in a unit-volume body $K\subset {\mathbb{R}}^n$ tends to infinity as $n\to \mathrm{\infty }$. However, for two small subsets of volume $\epsilon >0$ the situation is different. For unit-volume cubes and euclidean balls the largest distance is of order $\sqrt{-\ln \epsilon }$, for simplexes and hyperoctahedrons $-$ of order $-\ln \epsilon$, for ${\mathrm{\ell }}_p$ balls with $p\in [1;2]$ $-$ of order $(-\ln \epsilon {)}^{\frac{1}{p}}$. These estimates are not dependent on the dimensionality $n$. The goal of the paper is to study this phenomenon. Isoperimetric inequalities will play a key role in our approach.