High-Dimensional Menger-Type Curvatures - Part I: Geometric Multipoles and Multiscale Inequalities - Mathematics > Classical Analysis and ODEsReport as inadecuate




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Abstract: We define a discrete Menger-type curvature of d+2 points in a real separableHilbert space H by an appropriate scaling of the squared volume of thecorresponding d+1-simplex. We then form a continuous curvature of an Ahlforsd-regular measure on H by integrating the discrete curvature according to theproduct measure. The aim of this work, continued in a subsequent paper, is toestimate multiscale least squares approximations of such measures by theMenger-type curvature. More formally, we show that the continuous d-dimensionalMenger-type curvature is comparable to the ``Jones-type flatness-. The latterquantity adds up scaled errors of approximations of a measure by d-planes atdifferent scales and locations, and is commonly used to characterize uniformrectifiability. We thus obtain a characterization of uniform rectifiability byusing the Menger-type curvature. In the current paper part I we control thecontinuous Menger-type curvature of an Ahlfors d-regular measure by itsJones-type flatness.



Author: G. Lerman, J. T. Whitehouse

Source: https://arxiv.org/







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