Volumes spanned by k point configurations in ℝ^d

2021/02/03 by Galo, Belmiro, McDonald, Alex
#28A75 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2102.02323

Abstract

Given a k-point configuration x∈ (ℝ^d)^k, we consider the \binomkd-vector of volumes determined by choosing any d points of x. We prove that a compact set E⊂ \R^d determines a positive measure of such volume types if the Hausdorff dimension of E is greater than d-(d-1)/(2k-d). This generalizes results of Greenleaf, Iosevich, and Mourgoglou, Greenleaf, Iosevich, and Taylor, and the second listed author.

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