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

Abstract: Average distance between two points in a unit-volume body $K \subset \mathbb{R}^n$ tends to infinity as $n \to \infty$. However, for two small subsets of volume $\varepsilon > 0$ the situation is different. For unit-volume cubes and euclidean balls the largest distance is of order $\sqrt{-\ln \varepsilon}$, for simplexes and hyperoctahedrons $-$ of order $-\ln \varepsilon$, for $\ell_p$ balls with $p \in [1;2]$ $-$ of order $(-\ln \varepsilon)^{\frac{1}{p}}$. 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.