On the volumes of simplices determined by a subset of R^d

Abstract

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 \(k\)".