Volumes spanned by $k$ point configurations in $\mathbb{R}^d$
Abstract
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 $d\frac{d1}{2kd}$. This generalizes results of Greenleaf, Iosevich, and Mourgoglou, Greenleaf, Iosevich, and Taylor, and the second listed author.
 Publication:

arXiv eprints
 Pub Date:
 February 2021
 DOI:
 10.48550/arXiv.2102.02323
 arXiv:
 arXiv:2102.02323
 Bibcode:
 2021arXiv210202323G
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 28A75