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Menger curvatures and $C^{1,α}$ rectifiability of measures (1908.11471v2)
Published 29 Aug 2019 in math.MG and math.CA
Abstract: We further develop the relationship between $\beta$-numbers and discrete curvatures to provide a new proof that under weak density assumptions, finiteness of the pointwise discrete curvature $\operatorname{curv}{\alpha}_{\mu;2}(x,r)$ at $\mu$- a.e. $x \in \mathbb{R}{m}$ implies that $\mu$ is $C{1,\alpha}$ $n$-rectifiable.