On volumes of simplices in intermediate dimensions
Abstract
A variant of the Falconer distance problem asks for fixed k≥ 1 and d≥ k+1, how large does the Hausdorff dimension of a Borel set E⊂ℝd need to be to guarantee that there exist x0,…,xk∈ E such that Volk+1(x0,…,xk)(E) = { Volk+1(x0,…,xk,xk+1) : xk+1∈ E } has positive Lebesgue measure. Here Volk+1(x0,…,xk,xk+1) denotes the k+1-volume of the k+1 simplex formed by x0,…,xk,xk+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 ≤ d ≤ 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.
Related
- The Bézout inequality for mixed volumes characterizes simplices
- Angles and volumes of simplices, boxes, and crosspolytopes in constant curvature
- On the number of regular simplices in odd dimensions
- Morley Simplices in Higher Dimensions: Regularity, Reflections, and Counterexamples
- Sharp Projection Bounds for Intermediate Dimensions of Measures
BibTeX
@misc{montejo2026volumes,
title = {{On volumes of simplices in intermediate dimensions}},
author = {José Gaitan Montejo and Eyvindur Ari Palsson},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.22450},
eprint = {2605.22450},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.22450},
}
RIS
TY - UNPB
TI - On volumes of simplices in intermediate dimensions
AU - José Gaitan Montejo
AU - Eyvindur Ari Palsson
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2605.22450
UR - https://arxiv.org/abs/2605.22450
AB - A variant of the Falconer distance problem asks for fixed k≥ 1 and d≥ k+1, how large does the Hausdorff dimension of a Borel set E⊂ℝ^d need to be to guarantee that there exist x_0,…,x_k∈ E such that Vol_k+1^(x_0,…,x_k)(E) = { Vol_k+1(x_0,…,x_k,x_k+1) : x_k+1∈ E } has positive Lebesgue measure. Here Vol_k+1(x_0,…,x_k,x_k+1) denotes the k+1-volume of the k+1 simplex formed by x_0,…,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 ≤ d ≤ 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.
ER -