Source: Rev. Mat. Iberoamericana Volume 27, Number 2
(2011), 493-555.
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.
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