High-dimensional Menger-type curvatures. Part I: Geometric multipoles and multiscale inequalities

Abstract

We define discrete and continuous Menger-type curvatures. The discrete curvature scales the volume of a $(d+1)$-simplex in a real separable Hilbert space $H$, whereas the continuous curvature integrates the square of the discrete one according to products of a given measure (or its restriction to balls). The essence of this paper is to establish an upper bound on the continuous Menger-type curvature of an Ahlfors regular measure $\mu$ on $H$ in terms of the Jones-type flatness of $\mu$ (which adds up scaled errors of approximations of $\mu$ by $d$-planes at different scales and locations). As a consequence of this result we obtain that uniformly rectifiable measures satisfy a Carleson-type estimate in terms of the Menger-type curvature. 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.