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

TL;DR

This paper investigates how the maximum distance between small-volume subsets within high-dimensional bodies remains dimension-independent, revealing specific bounds for various geometric shapes using isoperimetric inequalities.

Contribution

The paper provides dimension-free estimates for distances between small-volume subsets in high-dimensional bodies, extending understanding of geometric properties across different shapes.

Findings

Maximum distances are independent of dimension for small-volume subsets.

Different shapes exhibit distinct logarithmic bounds on maximum distances.

Isoperimetric inequalities are key tools in deriving these estimates.

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.