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

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 mu 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 mu by the Menger-type curvature. We show that the continuous d-dimensional Menger-type curavture of mu is comparable to the "Jones-type flatness'' of mu. The latter quantity sums the scaled errors of approximations of mu by d-planes at different scales and locations, and is typically used to characterize the uniform rectifiability of mu.;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 mu restricted to a ball in terms of the Jones-type flatness of mu 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.