- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:05:46
Revista Matemática Iberoamericana
Rev. Mat. Iberoamericana
RMI
0213-2230
2235-0616
General
10.4171/RMI
http://www.ems-ph.org/doi/10.4171/RMI
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2012)
27
2011
2
High-dimensional Menger-type curvatures. Part I: Geometric multipoles and multiscale inequalities
Gilad
Lerman
University of Minnesota, MINNEAPOLIS, UNITED STATES
J. Tyler
Whitehouse
Vanderbilt University, NASHVILLE, UNITED STATES
Multiscale geometry, Ahlfors regular measure, uniform rectifiability, polar sine, Menger curvature, Menger-type curvature, least squares d-planes, recovering lowdimensional structures in high dimensions
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.
Measure and integration
Fourier analysis
Probability theory and stochastic processes
General
493
555
10.4171/RMI/645
http://www.ems-ph.org/doi/10.4171/RMI/645