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

TL;DR

This paper investigates the geometric properties of point configurations in Euclidean space, establishing conditions under which a set determines a positive measure of volume types based on its Hausdorff dimension.

Contribution

It generalizes previous results by establishing new dimension thresholds for sets to determine a positive measure of volume configurations.

Findings

Sets with Hausdorff dimension greater than a specific threshold determine positive measure of volume types.

Generalization of earlier results by Greenleaf, Iosevich, Mourgoglou, and Taylor.

Provides new bounds relating Hausdorff dimension to volume configuration measures.

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 $d-\frac{d-1}{2k-d}$. This generalizes results of Greenleaf, Iosevich, and Mourgoglou, Greenleaf, Iosevich, and Taylor, and the second listed author.