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

Given a $k$-point configuration $x\in (\mathbb{R}^d)^k$, we consider the $\binom{k}{d}$-vector of volumes determined by choosing any $d$ points of $x$. We prove that a compact set $E\subset \R^d$ 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.