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

TL;DR

This paper investigates the problem of estimating the distance between two small bodies within a high-dimensional unit volume ball, identifying the minimal surface area configuration for such bodies as dimension grows.

Contribution

It characterizes the minimal free surface area configuration of bodies of fixed volume inside a high-dimensional ball, revealing geometric properties in the limit as dimension increases.

Findings

Minimal surface area configuration is a half-space intersection with the ball.

Optimal bodies are symmetric and lie in a hyperplane through the center.

Results hold asymptotically as the dimension tends to infinity.

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.