64 research outputs found

    Minimal H\"older regularity implying finiteness of integral Menger curvature

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    We study two families of integral functionals indexed by a real number p>0p > 0. One family is defined for 1-dimensional curves in R3\R^3 and the other one is defined for mm-dimensional manifolds in Rn\R^n. These functionals are described as integrals of appropriate integrands (strongly related to the Menger curvature) raised to power pp. Given p>m(m+1)p > m(m+1) we prove that C1,αC^{1,\alpha} regularity of the set (a curve or a manifold), with α>α0=1−m(m+1)p\alpha > \alpha_0 = 1 - \frac{m(m+1)}p implies finiteness of both curvature functionals (m=1m=1 in the case of curves). We also show that α0\alpha_0 is optimal by constructing examples of C1,α0C^{1,\alpha_0} functions with graphs of infinite integral curvature

    High-Dimensional Menger-Type Curvatures-Part II: d-Separation and a Menagerie of Curvatures

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    This is the second of two papers wherein we estimate multiscale least squares approximations of certain measures by Menger-type curvatures. More specifically, we study an arbitrary d-regular measure on a real separable Hilbert space. The main result of the paper bounds the least squares error of approximation at any ball by an average of the discrete Menger-type curvature over certain simplices in in the ball. A consequent result bounds the Jones-type flatness by an integral of the discrete curvature over all simplices. The preceding paper provided the opposite inequalities. Furthermore, we demonstrate some other discrete curvatures for characterizing uniform rectifiability and additional continuous curvatures for characterizing special instances of the (p, q)-geometric property. We also show that a curvature suggested by Leger (Annals of Math, 149(3), p. 831-869, 1999) does not fit within our framework.Comment: 32 pages, no figure
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