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

This is the second of two papers wherein we estimate multiscale least squares approximations of certain measures by Menger-type curvatures (defined in Part I). More specifically, we study an arbitrary d-regular measure μ on a real separable Hilbert space, where d ∈ N. The main result of this paper bounds the least squares error of approximating μ at any ball B by an average of the discrete Menger-type curvature over certain simplices in B. A consequent result bounds the Jones-type flatness (which adds up least squares errors at different scales and locations) by an integral of the discrete curvature over all simplices in B. The preceding paper provided the opposite inequalities of these two results. Furthermore, we demonstrate a few 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 Léger (Annals of Math, 149(3), p. 831-869, 1999) does not fit within our framework. AMS Subject Classification (2000): 60D05, 49Q15, 42C99