Abstract It is a longstanding conjecture that given subset E of metric space, if has unit $\mathscr {H}^{\alpha }\llcorner E$ -density almost everywhere, then an $\alpha $ -rectifiable set. We prove this under the assumption ambient space homogeneous group with smooth-box norm.

We prove (without using Federer’s structure theorem) that a finite-mass flat chain over any coefficient group is rectifiable if and only if almost all of its 0-dimensional slices are rectifiable. This implies that every flat chain of finite mass and finite size is rectifiable. It also leads to a simple necessary and sufficient condition on the coefficient group in order for every finite-mass fl...

In this paper we prove the differentiability of Lipschitz maps X → V , where X is a complete metric measure space satisfying a doubling condition and a Poincaré inequality, and V denotes a Banach space with the Radon Nikodym Property (RNP). The proof depends on a new characterization of the differentiable structure on such metric measure spaces, in terms of directional derivatives in the direct...

We extend Federer’s coarea formula to mappings f belonging to the Sobolev class W (R;R), 1 ≤ m < n, p > m, and more generally, to mappings with gradient in the Lorentz space L(R). This is accomplished by showing that the graph of f in R is a Hausdorff n-rectifiable set.

We prove that there exists M > 0 such that for any closed rectifiable curve Γ in Hilbert space, almost every point in Γ is contained in a countable union of M chord-arc curves whose total length is no more than Ml(Γ). Mathematics Subject Classification (2000): 28A75

We study one dimensional sets (Hausdorff dimension) lying in a Hilbert space. The aim is to classify subsets of Hilbert spaces that are contained in a connected set of finite Hausdorff length. We do so by extending and improving results of Peter Jones and Kate Okikiolu for sets in Rd . Their results formed the basis of quantitative rectifiability in Rd . We prove a quantitative version of the f...

in this paper‎, ‎we mainly investigate how the generalized metrizability properties of the remainders affect the metrizability of rectifiable spaces‎, ‎and how the character of the remainders affects the character‎ ‎and the size of a rectifiable space‎. ‎some results in [a. v‎. ‎arhangel'skii and j‎. ‎van mill‎, ‎on topological groups with a first-countable remainder‎, ‎topology proc. 42 (2013...

A measure is 1-rectifiable if there is a countable union of finite length curves whose complement has zero measure. We characterize 1-rectifiable Radon measures μ in n-dimensional Euclidean space for all n ≥ 2 in terms of positivity of the lower density and finiteness of a geometric square function, which loosely speaking, records in an L gauge the extent to which μ admits approximate tangent l...