The boundness of distance between two sets of fixed volume inside the multidimensional ball or cube

TL;DR

This paper investigates the bounds on the distance between two fixed-volume bodies within high-dimensional spheres or cubes, revealing cases where these bounds are independent of the dimension.

Contribution

It provides new insights into the boundedness of distances between bodies of fixed volume in high-dimensional geometric spaces, especially for spheres and cubes.

Findings

Distances are bounded by functions depending only on volume, not dimension

Identifies specific cases where bounds are independent of the dimension

Extends understanding of geometric properties in high-dimensional spaces

Abstract

The problem of bounding of the distance between the two bodies of volume $\varepsilon$ located inside the $n$-dimensional body $B$ of unit volume where $n \to \infty$ is considered. In some cases such distances are bounded by function depends on $\varepsilon$ but not depends on $n$. We consider cases when $B$ is a sphere or a cube.