High-Dimensional Menger-Type Curvatures - Part I: Geometric Multipoles and Multiscale Inequalities

TL;DR

This paper introduces a new way to measure the curvature of measures in high-dimensional spaces using Menger-type curvature and shows it is comparable to Jones flatness, providing a new characterization of uniform rectifiability.

Contribution

It defines a discrete Menger-type curvature in Hilbert spaces and proves its equivalence to Jones flatness, linking geometric curvature to measure rectifiability.

Findings

Continuous Menger-type curvature is comparable to Jones flatness.

The work characterizes uniform rectifiability via Menger-type curvature.

Provides tools for multiscale geometric analysis in high dimensions.

Abstract

We define a discrete Menger-type curvature 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 d-regular measure on H by integrating the discrete curvature according to the product measure. The aim of this work, continued in a subsequent paper, is to estimate multiscale least squares approximations of such measures by the Menger-type curvature. More formally, we show that the continuous d-dimensional Menger-type curvature is comparable to the ``Jones-type flatness''. The latter quantity adds up scaled errors of approximations of a measure by d-planes at different scales and locations, and is commonly used to characterize uniform rectifiability. We thus obtain a characterization of uniform rectifiability by using the Menger-type curvature. In the current paper (part I) we control the continuous Menger-type curvature of an Ahlfors d-regular measure by its Jones-type flatness.