## 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 language | English (US) |
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Pages (from-to) | 325-360 |

Number of pages | 36 |

Journal | Constructive Approximation |

Volume | 30 |

Issue number | 3 |

DOIs | |

State | Published - Nov 2009 |

### Bibliographical note

Funding Information:This work has been supported by NSF grant #0612608

## Keywords

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