Higher order rectifiability of measures via averaged discrete curvatures

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 simplexes spanned by the parameters.