High-Dimensional Menger-Type Curvatures - Part I: Geometric Multipoles and Multiscale Inequalities
By Gilad Lerman, J. Tyler Whitehouse. 2008
50 citations recorded by the source corpus.
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.
Subfields: Applied Mathematics
View Paper: https://www.semanticscholar.org/paper/06e5a774013c8d5cf79ddf73a673e847207e6a50
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In Applied Mathematics
- Pawel Strzelecki
- H. von der Mosel
- Gilad Lerman
- J. Tyler Whitehouse
- Michael Struwe
- Changyou Wang
- Guangliang Chen
- Armin Schikorra
Citations in the corpus
- Integral Menger curvature for surfaces (2009, background): “In their pioneering work [15, 16] dealing with d-rectifiability and least square approximation of d-regular measures, G.”
- High-Dimensional Menger-Type Curvatures—Part II: d-Separation and a Menagerie of Curvatures (2008, methodology): “The methods of this paper and [ 10 ] extend to comparing e”
- Regularizing and self-avoidance effects of integral Menger curvature (2010, unknown): “A generalization of total Menger curvature M2(·) to higher-dimensional sets was recently initiated in [19,20].”