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

• Integrale Mengerkrümmung und Rektifizierbarkeit von n-dimensionalen Borel-Mengen im N-dimensionalen Euklidischen Raum

Meurer, Martin; von der Mosel, Heiko (Thesis advisor); Wagner, Alfred (Thesis advisor)

Aachen : Publikationsserver der RWTH Aachen University (2015)
Dissertation / PhD Thesis

Aachen, Techn. Hochsch., Diss., 2015

Abstract

In this thesis 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 β-numbers in terms of integral Menger curvature without assuming lower Ahlfors regularity, in contrast to the results of Lerman and Whitehouse.