Abstract. We provide a sufficient geometric condition for R n to be countably (µ, m) rectifiable of class C 1,α (using the terminology of Federer), where µ is a Radon measure having positive lower density and finite upper density µ almost everywhere. Our condition involves integrals of certain many-point interaction functions (discrete curvatures) which measure flatness of simplices spanned by the parameters.
Kolasiński REF provides a sufficient condition in terms of averaged discrete curvatures, similar to integral Menger curvatures, for a Radon measure with positive lower density and finite upper density to be C 1,α rectifiable.
Higher order rectifiability of measures via averaged discrete curvatures
Revista Matem\'atica Iberoamericana, Volume 33, Issue 3, 2017, pp. 861-884
Mathematics