High-dimensional Menger-type curvatures. I: Geometric multipoles and multiscale inequalities. (English) Zbl 1232.28007

Lerman, Gilad; Whitehouse, J. Tyler

Rev. Mat. Iberoam. 27, No. 2, 493-555 (2011).

The authors define discrete and continuous Menger-type curvatures. They establish an upper bound on the continuous Menger-type curvature of an Ahlfors regular measure \(\mu\). As a consequence, they prove that a uniformly rectifiable measure satisfy a Carleson-type estimate in terms of the Menger-type curvature.

Reviewer: D. R. Bell (Jacksonville)

MSC:

28A75	Length, area, volume, other geometric measure theory
42C99	Nontrigonometric harmonic analysis
60D05	Geometric probability and stochastic geometry

Keywords:

Multiscale geometry; Ahlfors regular measure

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