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May, 2011 High-dimensional Menger-type curvatures. Part I: Geometric multipoles and multiscale inequalities
Gilad Lerman , J. Tyler Whitehouse
Rev. Mat. Iberoamericana 27(2): 493-555 (May, 2011).


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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Gilad Lerman . J. Tyler Whitehouse . "High-dimensional Menger-type curvatures. Part I: Geometric multipoles and multiscale inequalities." Rev. Mat. Iberoamericana 27 (2) 493 - 555, May, 2011.


Published: May, 2011
First available in Project Euclid: 10 June 2011

zbMATH: 1232.28007
MathSciNet: MR2848529

Primary: 28A75 , 42C99 , 60D05

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

Rights: Copyright © 2011 Departamento de Matemáticas, Universidad Autónoma de Madrid

Vol.27 • No. 2 • May, 2011
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