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

Abdulamin Ismailov, Alexei Kanel-Belov, Fyodor Ivlev

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 $\epsilon >0$ t…