DIMENSION-FREE ESTIMATES ON DISTANCES BETWEEN SUBSETS OF VOLUME ε INSIDE A UNIT-VOLUME BODY

Abdulamin Ismailov, Alexei Kanel-Belov, Fyodor Ivlev

Research output: Contribution to journalReview articlepeer-review

Abstract

Average distance between two points in a unit-volume body K⊂ Rn tends to infinity as n→ ∞ . However, for two small subsets of volume ε> 0 , the situation is different. For unit-volume cubes and Euclidean balls, the largest distance is of order -lnε , for simplices and hyperoctahedra—of order - ln ε , for ℓp balls with p∈ [1 ; 2] —of order (-lnε)1p . 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.

Original languageEnglish
Pages (from-to)497-545
Number of pages49
JournalJournal of Mathematical Sciences
Volume271
Issue number4
DOIs
StatePublished - Apr 2023

Keywords

  • Concentration of measure
  • Isoperimetric inequalities

All Science Journal Classification (ASJC) codes

  • Applied Mathematics
  • Statistics and Probability
  • General Mathematics

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