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

Abstract

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 [19] 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 [20].

Higher order rectifiability of measures via averaged discrete curvatures

“…Consulting [LW09,LW11] and [Meu15, Theorem 5.6] which give estimates for β m Σ,2 in terms of the curvature energies one might acquire an impression that the integral in (6) and K l,p κ [Σ] might be mutually comparable if γ = 2/(m(m + 1)). However, by now it is not clear whether the results of [AT15] imply [Meu15] or vice versa. Certainly the two results were proven in parallel and using different techniques.…”

“…This is the famous theorem of David [Dav98] (see also Léger [Lég99]) proven in connection with the Vitushkin conjecture characterizing removable sets for bounded analytic functions. The result of Meurer [Meu15] constitutes a generalization of David's theorem and has been proven using Léger's approach. In this context, Farag [Far99] showed that there is no equivalent of the Menger-Melnikov curvature in higher dimensions which would relate to the Riesz transform in the way κ S relates to the Cauchy transform (cf.…”

“…Question (B) has been answered very recently in the thesis of Meurer [Meu15], who proved that if γ = 2/(m(m + 1)), p = m(m + 1), and ˆΣm+2 κ γ vol (q 0 , . .…”

Sufficient conditions for C^1,α parametrization and rectifiability

“…Several attempts have been made to prove similar analogues for sets (or measures) of dimension more than 1, see for instance [Paj96b,Paj96a]. Menger curvature was also introduced to attempt to characterize rectifiability (see, among others, [Lég99,LW11,LW09,KS13,BK12,Kol15,Meu18,Goe18,GG19]). Other approaches can be found in [Mer16,Del08,San19]).…”

Menger curvatures and $$\varvec{C^{1,\alpha }}$$ rectifiability of measures

“…Nonetheless, geometric arguments made with non-trivial adaptations from [Lég99] have since been used to characterize uniform rectifiability in all dimensions and codimensions in terms of Menger-type curvatures, [LW09,LW11]. A sufficient condition for rectifiability of sets in terms of higher dimensional Menger-type curvatures appears in [Meu18] and was extended to several characterizations of rectifiable measures under suitable density conditions [Goe18].…”