High-dimensional menger-type curvatures-Part II: D-separation and a menagerie of curvatures

Gilad Lerman, J. Tyler Whitehouse

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19 Scopus citations

Abstract

We estimate d-dimensional least squares approximations of an arbitrary d-regular measure μ via discrete curvatures of d + 2 variables. The main result bounds the least squares error of approximating μ (or its restrictions to balls) with a d-plane by an average of the discrete Menger-type curvature over a restricted set of simplices. Its proof is constructive and even suggests an algorithm for an approximate least squares d-plane. A consequent result bounds a multiscale error term (used for quantifying the approximation of μ with a sufficiently regular surface) by an integral of the discrete Menger-type curvature over all simplices. The preceding paper (part I) provided the opposite inequalities of these two results. This paper also demonstrates the use of a few other discrete curvatures which are different from the Menger-type curvature. Furthermore, it shows that a curvature suggested by Léger (Ann. Math. 149(3), pp. 831-869, 1999) does not fit within our framework.

Original languageEnglish (US)
Pages (from-to)325-360
Number of pages36
JournalConstructive Approximation
Volume30
Issue number3
DOIs
StatePublished - Nov 2009

Keywords

  • Ahlfors regular measure
  • Least squares d-planes
  • Menger-type curvature
  • Multiscale geometry
  • Polar sine
  • Uniform rectifiability

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