Volumes spanned by $k$ point configurations in ${\mathbb{R}}^d$

Given a $k$-point configuration $x\in ({\mathbb{R}}^d{)}^k$, we consider the $\left(\genfrac{}{}{0px}{}{k}{d}\right)$-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, Iosevich, and Mourgoglou, Greenleaf, Iosevich, and Taylor, and the second listed author.