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arXiv · 2301.13495

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

Abdulamin Ismailov·Alexei Kanel-Belov·Fyodor Ivlev
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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.

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Abdulamin Ismailov, Alexei Kanel-Belov, Fyodor Ivlev. 2023-01-31. Dimension-free estimates on distances between subsets of volume $\varepsilon$ inside a unit-volume body. https://doi.org/10.1007/s10958-023-06622-8

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