Curvature measures

Curvature measures and fractals

Curvature measures and fractals

Curvature measures are an important tool in geometric measure theory and other fields of mathematics for describing the geometry of sets in Euclidean space. But the ‘classical’ concepts of curvature are not directly applicable to fractal sets. We try to bridge this gap between geometric measure theory and fractal geometry by introducing a notion of curvature for fractals. For compact sets F ⊆ R...

Stability of Curvature Measures

We address the problem of curvature estimation from sampled compact sets. The main contribution is a stability result: we show that the gaussian, mean or anisotropic curvature measures of the offset of a compact set K with positive μ-reach can be estimated by the same curvature measures of the offset of a compact set K’ close to K in the Hausdorff sense. We show how these curvature measures can...

Kurtosis and Curvature Measures for Nonlinear Regression Models

An expression for the second-order approximation to the kurtosis associated with the least squares estimate of an individual parameter in a nonlinear regression model is derived, and connections between this and various other measures of curvature are made. Furthermore a means of predicting the reliability of the commonly-used Wald confidence intervals for individual model parameters, based on ...

The Christoffel-minkowski Problem Iii: Existence Problem for Curvature Measures

Curvature measure is one of the basic notion in the theory of convex bodies. Together with surface area measures, they play fundamental roles in the study of convex bodies. They are closely related to the differential geometry and integral geometry of convex hypersurfaces. Let Ω is a bounded convex body in R with C2 boundary M , the corresponding curvature measures and surface area measures of ...