Estimation of the distance between two bodies inside an $n$-dimensional ball of unit volume

Abstract

We consider the problem of estimating the distance between two bodies of volume $\varepsilon$ located inside a $n$-dimensional ball $U$ of unit volume for $n\to\infty$. Let $A$ be a closed set with a smooth boundary of the volume $\varepsilon$ ($0 \leq \varepsilon \leq 1/2$) inside a $n$-dimensional ball $U$ of unit volume that implements among all the sets of volume $\varepsilon$ is a set with the smallest possible free surface area, lying in one half-space with respect to a certain hyperplane that passes through the center of the ball. Then $A$ has the same free surface area as the set representing the intersection of a ball perpendicular to $U$ and the ball $U$ itself.