In his 1990 paper, Jones characterized subsets of rectifiable curves in R2 via a multiscale sum β-numbers, which measure how far given set deviates from straight line at each scale and location. This characterization was extended by Okikiolu to Rn Schul Hilbert space. Recently, there has been some interest characterizing higher dimensional surfaces Rn. Using variant Jones' β-number introduced A...

We prove that 2 dimensional Integral currents (i.e. integer multiplicity 2 dimensional rectifiable currents) which are almost complex cycles in an almost complex manifold admitting locally a compatible symplectic form are smooth submanifolds aside from isolated points and therefore are J−holomorphic curves.

We prove that any Kantorovich potential for the cost function c = d/2 on a Riemannian manifold (M, g) is locally semiconvex in the “region of interest”, without any compactness assumption on M , nor any assumption on its curvature. Such a region of interest is of full μ-measure as soon as the starting measure μ does not charge n − 1-dimensional rectifiable sets.

In 1955, Martin Kneser showed that the Minkowski content of a compact p-rectifiable subset M of Rn is equal to its p-Hausdorff measure: lim t→0,t>0 Ln ` B(M, t) ́ α(n− p) tn−p = H(M). We extend his result to the reachable sets of a linear control system

Given a lower semicontinuous function f : Rn → R ∪ {+∞}, we prove that the set of points of Rn where the lower Dini subdifferential has convex dimension k is countably (n − k)-rectifiable. In this way, we extend a theorem of Benoist(see [1, Theorem 3.3]), and as a corollary we obtain a classical result concerning the singular set of locally semiconcave functions.

If E ⊂ C is a set with finite length and finite curvature, then E is rectifiable. This fact, proved by David and Léger in 1999, is one of the basic ingredients for the proof of Vitushkin’s conjecture. In this paper we give another different proof of this result.

in which P(xy) =x; and it is to be noted that here C does not need to be rectifiable. In the present paper a definition of relative content is given which makes it possible to prove that if P and Px are subject to certain conditions, the content of K, relative to a certain non-additive function of rectangles derived from P, exists equal to the double integral on the left of (1) and also equal t...

We prove that every polynomially convex arc is contained in a simple closed curve. also establish results about polynomial hulls of arcs and curves are locally rectifiable outside subset.

We identify two sufficient conditions for locally finite Borel measures on R to give full mass to a countable family of Lipschitz images of R. The first condition, extending a prior result of Pajot, is a sufficient test in terms of L affine approximability for a locally finite Borel measure μ on R satisfying the global regularity hypothesis lim sup r↓0 μ(B(x, r))/r <∞ at μ-a.e. x ∈ R to be m-re...

In mathematics curves are defined as the images of continuous real functions defined on closed intervals and these continuous functions are called parameterizations of the corresponding curves. If only simple curves of finite lengths are considered, then parameterizations can be restricted to the injective continuous functions or even to the continuous length-normalized parameterizations. In ad...