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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.
We design a fast algorithm that computes, for a given linear differential operator with coefficients in $Z[x ]$, all the characteristic polynomials of its p-curvatures, for all primes $p < N$ , in asymptotically quasi-linear bit complexity in N. We d
Let $(M,g)$ be a complete three dimensional Riemannian manifold with boundary $partial M$. Given smooth functions $K(x)>0$ and $c(x)$ defined on $M$ and $partial M$, respectively, it is natural to ask whether there exist metrics conformal to $g$ so t
We investigate the intrinsic and extrinsic curvatures of certain hypersurfaces in the thermodynamic geometry of a physical system and show that they contain useful thermodynamic information. For an anti-Reissner-Nordstr{o}m-(A)de Sitter black hole (P
We give criteria for which a principal curvature becomes a bounded $C^infty$-function at non-degenerate singular points of wave fronts by using geometric invariants. As applications, we study singularities of parallel surfaces and extended distance s
The {em focal curve} of an immersed smooth curve $gamma:smapsto gamma(s)$, in Euclidean space $R^{m+1}$, consists of the centres of its osculating hyperspheres. The focal curve may be parametrised in terms of the Frenet frame of $gamma$ (${bf t},{bf