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This is the second of two papers wherein we estimate multiscale least squaresapproximations of certain measures by Menger-type curvatures. Morespecifically, we study an arbitrary d-regular measure on a real separableHilbert space. The main result of the paper bounds the least squares error ofapproximation at any ball by an average of the discrete Menger-type curvatureover certain simplices in in the ball. A consequent result bounds theJones-type flatness by an integral of the discrete curvature over allsimplices. The preceding paper provided the opposite inequalities. Furthermore,we demonstrate some other discrete curvatures for characterizing uniformrectifiability and additional continuous curvatures for characterizing specialinstances of the (p, q)-geometric property. We also show that a curvaturesuggested by Leger (Annals of Math, 149(3), p. 831-869, 1999) does not fitwithin our framework.

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