We define discrete and continuous Menger-type curvatures. The discrete curvature scales the volume of a -simplex in a real separable Hilbert space , 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 on in terms of the Jones-type flatness of (which adds up scaled errors of approximations of by -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.
Cite this article
Gilad Lerman, J. Tyler Whitehouse, High-dimensional Menger-type curvatures. Part I: Geometric multipoles and multiscale inequalities. Rev. Mat. Iberoam. 27 (2011), no. 2, pp. 493–555DOI 10.4171/RMI/645