  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.
