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

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.

32 pages, no figures

References in corpus (2)

• Foundations of a Multi-way Spectral Clustering Framework for Hybrid Linear Modeling
• On d-dimensional d-Semimetrics and Simplex-Type Inequalities for High-Dimensional Sine Functions

Cited by in corpus (8)

• Foundations of a Multi-way Spectral Clustering Framework for Hybrid Linear Modeling
• Integral Menger curvature for surfaces
• Characterizing $W^{2,p}$~submanifolds by $p$-integrability of global curvatures
• Sharp Boundedness and Regularizing effects of the integral Menger curvature for submanifolds
• Geometric Sobolev-like embedding using high-dimensional Menger-like curvature
• Minimal Hölder regularity implying finiteness of integral Menger curvature
• Least squares approximations of measures via geometric condition numbers
• Local scales on curves and surfaces