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

Abdulamin Ismailov, Alexei Kanel-Belov, Fyodor Ivlev

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.

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

Abstract

Average distance between two points in a unit-volume body tends to infinity as . However, for two small subsets of volume the situation is different. For unit-volume cubes and euclidean balls the largest distance is of order , for simplexes and hyperoctahedrons of order , for balls with of order . These estimates are not dependent on the dimensionality . The goal of the paper is to study this phenomenon. Isoperimetric inequalities will play a key role in our approach.
Paper Structure (14 sections, 40 theorems, 388 equations, 2 figures)

This paper contains 14 sections, 40 theorems, 388 equations, 2 figures.

Table of Contents

  1. Introduction.
  2. Preliminaries.
  3. Euclidean balls.
  4. Unit cubes.
  5. Simplexes and $\ell_p$-balls.
  6. Simplexes.
  7. $\ell_p$-balls.
  8. Lower bounds.
  9. Discrete isoperimetric problem.
  10. Conclusions.
  11. Asymptotic behavior of $\Phi^{-1}$.
  12. Function $x (-\log x)^{1 - \frac{1}{p}}$ is increasing on $(0; \frac{1}{2}]$.
  13. Functions $V_n(x)$ and $S_n(x)$ in limit.
  14. Average distance.

Key Result

Theorem 1

When $K_n$ are the unit-volume euclidean balls

Figures (2)

  • Figure 1: The right triangle described above
  • Figure :

Theorems & Definitions (67)

  • Theorem : Theorem \ref{['ball_exact']}.
  • Theorem : Theorems \ref{['cube_upper']} and \ref{['cubes']}.
  • Theorem : Theorems \ref{['simplex_upper']} and \ref{['simplex']}.
  • Theorem : Theorems \ref{['ell_p_upper']} and \ref{['l_p_balls']}.
  • Theorem : Theorem \ref{['discrete_cube']}.
  • Theorem : Theorem \ref{['general_sup']}.
  • Definition 1
  • Definition 2
  • Lemma 2.1
  • proof
  • ...and 57 more