On the volumes of simplices determined by a subset of Rd

Abstract

We prove that for 1 ≤ k < d, if E is a Borel subset of Rd 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”.