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Integral Menger curvature and rectifiability of $ n$-dimensional Borel sets in Euclidean $ N$-space


Author: Martin Meurer
Journal: Trans. Amer. Math. Soc. 370 (2018), 1185-1250
MSC (2010): Primary 28A75; Secondary 28A80, 42B20
DOI: https://doi.org/10.1090/tran/7011
Published electronically: August 15, 2017
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Abstract: In this paper 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 Léger'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 [Constr. Approx. 30 (2009), 325-360].


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Additional Information

Martin Meurer
Affiliation: Institut für Mathematik, RWTH Aachen University, Templergraben 55, D-52062 Aachen, Germany
Email: meurer@instmath.rwth-aachen.de

DOI: https://doi.org/10.1090/tran/7011
Keywords: Geometric measure theory, Menger curvature, rectifiability, $\beta$-numbers
Received by editor(s): November 16, 2015
Received by editor(s) in revised form: May 22, 2016, and June 3, 2016
Published electronically: August 15, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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