Generalized sines, multiway curvatures, and the multiscale geometry of d-Regular measures.

Abstract

We define discrete Menger-type curvatures of d + 2 points in a real separable Hilbert space H by an appropriate scaling of the squared volume of the corresponding ( d + 1)-simplex. We then form a continuous curvature of an Ahlfors regular measure μ on H by integrating the discrete curvature according to products of mu (or its restriction to balls). The essence of this work is estimating multiscale least squares approsimations of μ by the Menger-type curvature. We show that the continuous d -dimensional Menger-type curavture of μ is comparable to the "Jones-type flatness'' of μ. The latter quantity sums the scaled errors of approximations of μ by d -planes at different scales and locations, and is typically used to characterize the uniform rectifiability of μ.

This work is divided into three basic parts, with the first part dealing with various geometric inequalities for the d -dimensional polar sine and hyper sine functions, which are higher-dimensional generalizations of the ordinary trigonometric sine function of an angle. The polar sine function is then used to formulate the Menger-type curvature in terms of a scaled volume. The second two parts use these geometric inequalities and their interaction with the geometry of d -regular measures to establish both an upper bound and a lower bound for the Menger-type curvature of μ restricted to a ball in terms of the Jones-type flatness of μ restricted to a ball. In addition to the Menger-type curvatures, we give a brief exploration of various other curvatures in the context of comparisons to the the Jones-type flatness and their use in the context of uniform rectifiability.