On the volumes of simplices determined by a subset of R^d
Keywords:
Patterns, configurations, simplices, volumes, Hausdorff dimension, slices, projectionsAbstract
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 \(k\)".How to Cite
Shmerkin, P., & Yavicoli, A. (2025). On the volumes of simplices determined by a subset of R^d. Annales Fennici Mathematici, 50(1), 97–108. https://doi.org/10.54330/afm.159807
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