On the volumes of simplices determined by a subset of $\mathbb{R}^d$
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2023
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866912297963749376 |
|---|---|
| author | Shmerkin, Pablo Yavicoli, Alexia |
| author_facet | Shmerkin, Pablo Yavicoli, Alexia |
| contents | We prove that for $1\le k<d$, if $E$ is a Borel subset of $\mathbb{R}^d$ of Hausdorff dimension strictly larger than $k$, the set of $(k+1)$-volumes determined by $k+2$ points in $E$ has positive one-dimensional Lebesgue measure. In the case $k=d-1$, we obtain an essentially sharp lower bound on the dimension of the set of tuples in $E$ generating a given volume. We also establish a finer version of the classical slicing theorem of Marstrand-Mattila in terms of dimension functions, and use it to extend our results to sets of ``dimension logarithmically larger than $k$''. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_17855 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the volumes of simplices determined by a subset of $\mathbb{R}^d$ Shmerkin, Pablo Yavicoli, Alexia Classical Analysis and ODEs Combinatorics 28A12, 28A78, 28A80 We prove that for $1\le k<d$, if $E$ is a Borel subset of $\mathbb{R}^d$ of Hausdorff dimension strictly larger than $k$, the set of $(k+1)$-volumes determined by $k+2$ points in $E$ has positive one-dimensional Lebesgue measure. In the case $k=d-1$, we obtain an essentially sharp lower bound on the dimension of the set of tuples in $E$ generating a given volume. We also establish a finer version of the classical slicing theorem of Marstrand-Mattila in terms of dimension functions, and use it to extend our results to sets of ``dimension logarithmically larger than $k$''. |
| title | On the volumes of simplices determined by a subset of $\mathbb{R}^d$ |
| topic | Classical Analysis and ODEs Combinatorics 28A12, 28A78, 28A80 |
| url | https://arxiv.org/abs/2310.17855 |