Estimating discrete curvatures in terms of beta numbers

TL;DR

This paper develops a method to estimate the integrated discrete curvature of a Radon measure using beta numbers, connecting different variants and providing a partial converse to a recent theorem, advancing geometric measure theory.

Contribution

It introduces estimates of discrete curvature in terms of beta numbers and relates centered and non-centered variants, extending previous theoretical results.

Findings

Established bounds for integrated discrete curvature using beta numbers.

Related centered and non-centered beta numbers through integral estimates.

Derived a partial converse to Meurer's theorem using recent results of Tolsa.

Abstract

For an arbitrary Radon measure $\mu$ we estimate the integrated discrete curvature of $\mu$ in terms of its centred variant of Jones' beta numbers. We farther relate integrals of centred and non-centred beta numbers. As a corollary, employing the recent result of Tolsa [Calc. Var. PDE, 2015], we obtain a partial converse of the theorem of Meurer [arXiv:1510.04523].