High-Dimensional Menger-Type Curvatures - Part I: Geometric Multipoles and Multiscale Inequalities

We define discrete Menger-type curvature of d+2 points in a real separable Hilbert space H by an appropriate scaling of the squared volume of the corresponding (d+1)-simplex. We then form a continuous curvature of an Ahlfors regular measure μ on H by integrating the discrete curvature according to products of μ (or its restriction to balls). The essence of this work, which continues in a subsequent paper, is to estimate multiscale least squares approximations of μ by the Menger-type curvature. More formally, we show that the continuous d-dimensional Menger-type curvature of μ is comparable to the “Jones-type flatness” of μ. The latter quantity adds up scaled errors of approximations of μ by d-planes at different scales and locations, and is commonly used to characterize uniform rectifiability. We thus obtain a characterization of uniform rectifiability by using the Menger-type curvature. In the current paper (part I) we control the continuous Menger-type curvature of an Ahlfors regular measure by its Jones-type flatness. Our strategy combines discrete and integral multiscale inequalities for the polar sine with the “geometric multipoles” construction, which is a multiway analog of the well-known method of fast multipoles. AMS Subject Classification (2000): 49Q15, 42C99, 60D05