JournalsrmiVol. 27, No. 2pp. 493–555

High-dimensional Menger-type curvatures. Part I: Geometric multipoles and multiscale inequalities

  • Gilad Lerman

    University of Minnesota, Minneapolis, USA
  • J. Tyler Whitehouse

    Vanderbilt University, Nashville, USA
High-dimensional Menger-type curvatures. Part I: Geometric multipoles and multiscale inequalities cover
Download PDF

Abstract

We define discrete and continuous Menger-type curvatures. The discrete curvature scales the volume of a (d+1)(d+1)-simplex in a real separable Hilbert space HH, whereas the continuous curvature integrates the square of the discrete one according to products of a given measure (or its restriction to balls). The essence of this paper is to establish an upper bound on the continuous Menger-type curvature of an Ahlfors regular measure μ\mu on HH in terms of the Jones-type flatness of μ\mu (which adds up scaled errors of approximations of μ\mu by dd-planes at different scales and locations). As a consequence of this result we obtain that uniformly rectifiable measures satisfy a Carleson-type estimate in terms of the Menger-type curvature. Our strategy combines discrete and integral multiscale inequalities for the polar sine with the "geometric multipoles" construction, which is a multiway analog of the well-known method of fast multipoles.

Cite this article

Gilad Lerman, J. Tyler Whitehouse, High-dimensional Menger-type curvatures. Part I: Geometric multipoles and multiscale inequalities. Rev. Mat. Iberoam. 27 (2011), no. 2, pp. 493–555

DOI 10.4171/RMI/645