Mathematics – Classical Analysis and ODEs
Scientific paper
2008-05-09
Rev. Mat. Iberoamericana 27 (2011), no. 2, 493-555
Mathematics
Classical Analysis and ODEs
47 pages, 13 figures. Minor revisions and the inclusion of figures
Scientific paper
We define a discrete Menger-type curvature of d+2 points in a real separable Hilbert space H by an appropriate scaling of the squared volume of the corresponding (d+1)-simplex. We then form a continuous curvature of an Ahlfors d-regular measure on H by integrating the discrete curvature according to the product measure. The aim of this work, continued in a subsequent paper, is to estimate multiscale least squares approximations of such measures by the Menger-type curvature. More formally, we show that the continuous d-dimensional Menger-type curvature is comparable to the ``Jones-type flatness''. The latter quantity adds up scaled errors of approximations of a measure by d-planes at different scales and locations, and is commonly used to characterize uniform rectifiability. We thus obtain a characterization of uniform rectifiability by using the Menger-type curvature. In the current paper (part I) we control the continuous Menger-type curvature of an Ahlfors d-regular measure by its Jones-type flatness.
Lerman Galia
Whitehouse Jonathan Tyler
No associations
LandOfFree
High-Dimensional Menger-Type Curvatures - Part I: Geometric Multipoles and Multiscale Inequalities does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with High-Dimensional Menger-Type Curvatures - Part I: Geometric Multipoles and Multiscale Inequalities, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and High-Dimensional Menger-Type Curvatures - Part I: Geometric Multipoles and Multiscale Inequalities will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-188546