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arXiv · 1510.04523

Integral Menger Curvature and Rectifiability of $n$-dimensional Borel sets in Euclidean $N$-space

Martin Meurer
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Abstract

In this work we show that an $n$-dimensional Borel set in Euclidean $N$-space with finite integral Menger curvature is $n$-rectifiable, meaning that it can be covered by countably many images of Lipschitz continuous functions up to a null set in the sense of Hausdorff measure. This generalises L\'{e}ger's rectifiability result for one-dimensional sets to arbitrary dimension and co-dimension. In addition, we characterise possible integrands and discuss examples known from the literature. Intermediate results of independent interest include upper bounds of different versions of P. Jones's $\beta$-numbers in terms of integral Menger curvature without assuming lower Ahlfors regularity, in contrast to the results of Lerman and Whitehouse.

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Martin Meurer. 2015-10-15. Integral Menger Curvature and Rectifiability of $n$-dimensional Borel sets in Euclidean $N$-space. https://arxiv.org/abs/1510.04523

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