Volumes Spanned by k-Point Configurations in $${\mathbb {R}}^d$$

Abstract

Given a k-point configuration $$x\in ({\mathbb {R}}^d)^k$$ (where $$k\ge d$$ ), we consider the $$\left( {\begin{array}{c}k\\ d\end{array}}\right) $$ -vector of volumes determined by choosing any d points of x. We prove that a compact set $$E\subset {\mathbb {R}}^d$$ determines a positive measure of such volume types if the Hausdorff dimension of E is greater than $$d-\frac{d-1}{2k-d}$$ . This generalizes results of Greenleaf et al. (Forum Math 27(1):635–646, 2015) and (On k-point configuration sets with nonempty interior, https://arxiv.org/abs/2005.10796 , 2020) and the McDonald (in: Proceedings of the AMS, https://arxiv.org/abs/2008.13720 ).