On volumes of simplices in intermediate dimensions
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917520051535872 |
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| author | Montejo, José Gaitan Palsson, Eyvindur Ari |
| author_facet | Montejo, José Gaitan Palsson, Eyvindur Ari |
| contents | A variant of the Falconer distance problem asks for fixed $k\geq 1$ and $d\geq k+1$, how large does the Hausdorff dimension of a Borel set $E\subset\mathbb{R}^d$ need to be to guarantee that there exist $x_0,\ldots,x_{k}\in E$ such that $\text{Vol}_{k+1}^{(x_0,\ldots,x_{k})}(E) = \lbrace \text{Vol}_{k+1}(x_0,\ldots,x_{k},x_{k+1}) : x_{k+1}\in E \rbrace$ has positive Lebesgue measure. Here $\text{Vol}_{k+1}(x_0,\ldots,x_{k},x_{k+1})$ denotes the $k+1$-volume of the $k+1$ simplex formed by $x_0,\ldots,x_{k},x_{k+1}$. Recently, Shmerkin and Yavicoli established a sharp dimensional threshold $k$ in the case when $d=k+1$. In this paper we extend their result to $k+1 \leq d \leq 2k$ and obtain a non-trivial dimensional threshold $d-k$ when $d>2k$.
The result is motivated by ideas from Shmerkin and Yavicoli. A crucial part of the argument is an application of work by Bright, Ortiz and Zakharov on a continuum Beck-type theorem for hyperplanes as well as classic results of Marstrand on projections and slicing theorems. In addition, we investigate a more elementary approach under a condition called the Fubini property for Hausdorff dimension as introduced in the work of Héra, Keleti and Máthé. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_22450 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On volumes of simplices in intermediate dimensions Montejo, José Gaitan Palsson, Eyvindur Ari Classical Analysis and ODEs 28A12, 28A78, 28A80 A variant of the Falconer distance problem asks for fixed $k\geq 1$ and $d\geq k+1$, how large does the Hausdorff dimension of a Borel set $E\subset\mathbb{R}^d$ need to be to guarantee that there exist $x_0,\ldots,x_{k}\in E$ such that $\text{Vol}_{k+1}^{(x_0,\ldots,x_{k})}(E) = \lbrace \text{Vol}_{k+1}(x_0,\ldots,x_{k},x_{k+1}) : x_{k+1}\in E \rbrace$ has positive Lebesgue measure. Here $\text{Vol}_{k+1}(x_0,\ldots,x_{k},x_{k+1})$ denotes the $k+1$-volume of the $k+1$ simplex formed by $x_0,\ldots,x_{k},x_{k+1}$. Recently, Shmerkin and Yavicoli established a sharp dimensional threshold $k$ in the case when $d=k+1$. In this paper we extend their result to $k+1 \leq d \leq 2k$ and obtain a non-trivial dimensional threshold $d-k$ when $d>2k$. The result is motivated by ideas from Shmerkin and Yavicoli. A crucial part of the argument is an application of work by Bright, Ortiz and Zakharov on a continuum Beck-type theorem for hyperplanes as well as classic results of Marstrand on projections and slicing theorems. In addition, we investigate a more elementary approach under a condition called the Fubini property for Hausdorff dimension as introduced in the work of Héra, Keleti and Máthé. |
| title | On volumes of simplices in intermediate dimensions |
| topic | Classical Analysis and ODEs 28A12, 28A78, 28A80 |
| url | https://arxiv.org/abs/2605.22450 |