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

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 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.

We added missing exponents in the definition of a proper integrand (def 3.1)

References in corpus (2)

• Geometric Sobolev-like embedding using high-dimensional Menger-like curvature
• For which positive $p$ is the integral Menger curvature ${\mathcal{M}}_p$ finite for all simple polygons?

Cited by in corpus (1)

• Estimating discrete curvatures in terms of beta numbers