We prove that for $1\le k, 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 $k$".
1
The University of British Columbia, Department of Mathematics
Pablo Shmerkin; Alexia Yavicoli. On the volumes of simplices determined by a subset of R^d. Annales Fennici Mathematici, Tome 50 (2025) no. 1, p. 97–108. doi: 10.54330/afm.159807
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author = {Pablo Shmerkin and Alexia Yavicoli},
title = {On the volumes of simplices determined by a subset of {R^d}},
journal = {Annales Fennici Mathematici},
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