TY - JOUR

T1 - High-dimensional menger-type curvatures-Part II

T2 - D-separation and a menagerie of curvatures

AU - Lerman, Gilad

AU - Whitehouse, J. Tyler

PY - 2009/11

Y1 - 2009/11

N2 - 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.

AB - 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.

KW - Ahlfors regular measure

KW - Least squares d-planes

KW - Menger-type curvature

KW - Multiscale geometry

KW - Polar sine

KW - Uniform rectifiability

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U2 - 10.1007/s00365-009-9073-z

DO - 10.1007/s00365-009-9073-z

M3 - Article

AN - SCOPUS:70449527779

VL - 30

SP - 325

EP - 360

JO - Constructive Approximation

JF - Constructive Approximation

SN - 0176-4276

IS - 3

ER -