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

TL;DR

This paper proves that n-dimensional sets with finite integral Menger curvature in Euclidean space are n-rectifiable, extending previous results from one-dimensional sets to higher dimensions and co-dimensions.

Contribution

It generalizes Légers rectifiability result to higher dimensions, characterizes integrands, and relates Menger curvature to Jones's β-numbers without regularity assumptions.

Findings

Finite integral Menger curvature implies n-rectifiability.

Provides bounds for Jones's β-numbers via Menger curvature.

Extends rectifiability theory to arbitrary dimensions and co-dimensions.

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.