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 language | English |
|---|---|
| Pages (from-to) | 497-545 |
| Number of pages | 49 |
| Journal | Journal of Mathematical Sciences (United States) |
| Volume | 271 |
| Issue number | 4 |
| DOIs | |
| State | Published - 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).
| Funders | Funder number |
|---|---|
| Nikolay Antonovich Bobylev | |
| Russian Science Foundation | 22-11-00177 |
Keywords
- Concentration of measure
- Isoperimetric inequalities
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