We define discrete and continuous Menger-type curvatures. Thediscrete curvature scales the volume of a $(d+1)$-simplex in a realseparable Hilbert space $H$, whereas the continuous curvatureintegrates the square of the discrete one according to products of agiven measure (or its restriction to balls). The essence of thispaper is to establish an upper bound on the continuous Menger-typecurvature of an Ahlfors regular measure $\mu$ on $H$ in terms ofthe Jones-type flatness of $\mu$ (which adds up scaled errors ofapproximations of $\mu$ by $d$-planes at different scales andlocations). As a consequence of this result we obtain that uniformlyrectifiable measures satisfy a Carleson-type estimate in terms ofthe 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.
"High-dimensional Menger-type curvatures. Part I: Geometric multipoles and multiscale inequalities." Rev. Mat. Iberoamericana 27 (2) 493 - 555, May, 2011.