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

Bibliographical note

Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

Funding

This work was supported by Russian Science Foundation Grant No. 22-11-00177 https://rscf.ru/project/22-11-00177/ . We thank Sasha Sodin and Sergey Bobkov for their valuable feedback. We also thank Alexey Karapetyants for his help and advice. Our work was inspired by a problem posed by Nikolay Antonovich Bobylev, which also led to a project at the 34th Summer Conference of the International Mathematical Tournament of Towns (https://www.turgor.ru/lktg/2022/5/index.html).

FundersFunder number
Nikolay Antonovich Bobylev
Russian Science Foundation22-11-00177

    Keywords

    • Concentration of measure
    • Isoperimetric inequalities

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