High-Dimensional Menger-Type Curvatures-Part II: d-Separation and a Menagerie of Curvatures

TL;DR

This paper develops bounds for multiscale least squares approximation errors of measures in Hilbert spaces using Menger-type curvatures, linking discrete and continuous curvatures to geometric properties like rectifiability.

Contribution

It introduces new bounds relating least squares errors to Menger-type curvatures and explores various discrete and continuous curvatures for characterizing geometric properties.

Findings

Bounds the least squares error by average discrete Menger-type curvature.

Establishes bounds on Jones-type flatness via integral of discrete curvature.

Shows certain Leger curvature does not fit within the proposed framework.

Abstract

This is the second of two papers wherein we estimate multiscale least squares approximations of certain measures by Menger-type curvatures. More specifically, we study an arbitrary d-regular measure on a real separable Hilbert space. The main result of the paper bounds the least squares error of approximation at any ball by an average of the discrete Menger-type curvature over certain simplices in in the ball. A consequent result bounds the Jones-type flatness by an integral of the discrete curvature over all simplices. The preceding paper provided the opposite inequalities. Furthermore, we demonstrate some other discrete curvatures for characterizing uniform rectifiability and additional continuous curvatures for characterizing special instances of the (p, q)-geometric property. We also show that a curvature suggested by Leger (Annals of Math, 149(3), p. 831-869, 1999) does not fit within our framework.