Revista Matemática Iberoamericana

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Volume 27, Issue 2, 2011, pp. 493–555
DOI: 10.4171/RMI/645

Published online: 2011-08-31

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

Gilad Lerman[1] and J. Tyler Whitehouse[2]

(1) University of Minnesota, Minneapolis, USA
(2) Vanderbilt University, Nashville, USA

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.

Keywords: Multiscale geometry, Ahlfors regular measure, uniform rectifiability, polar sine, Menger curvature, Menger-type curvature, least squares d-planes, recovering lowdimensional structures in high dimensions

Lerman Gilad, Whitehouse J. Tyler: High-dimensional Menger-type curvatures. Part I: Geometric multipoles and multiscale inequalities. Rev. Mat. Iberoamericana 27 (2011), 493-555. doi: 10.4171/RMI/645