Revista Matemática Iberoamericana


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Volume 33, Issue 3, 2017, pp. 861–884
DOI: 10.4171/RMI/958

Published online: 2017-10-02

Higher order rectifiability of measures via averaged discrete curvatures

Sławomir Kolasiński[1]

(1) University of Warsaw, Poland

We provide a sufficient geometric condition for $\mathbb R^n$ to be countably $(\mu,m)$ rectifiable of class $\mathcal C^{1,\alpha}$ (using the terminology of Federer), where $\mu$ is a Radon measure having positive lower density and finite upper density $\mu$ almost everywhere. Our condition involves integrals of certain many-point interaction functions (discrete curvatures) which measure flatness of simplexes spanned by the parameters.

Keywords: Rectifiability of higher order, Lusin property for sets, Menger curvature

Kolasiński Sławomir: Higher order rectifiability of measures via averaged discrete curvatures. Rev. Mat. Iberoamericana 33 (2017), 861-884. doi: 10.4171/RMI/958