Aim of this paper is a finer analysis of the group of flat chains with coefficients in Zp introduced in [7], by taking quotients of the group of integer rectifiable currents, along the lines of [27, 15]. We investigate the typical questions of the theory of currents, namely rectifiability of the measure-theoretic support and boundary rectifiability. Our main result can also be interpreted as a ...

I. Let C be a closed rectifiable curve or an open rectifiable arc in the 77-dimensional Euclidean space En. Let C, V(C) and L(C) denote respectively the closed convex hull of C, the volume of C and the length of C. Let (c, n), or (0, n), stand for the problem of maximizing V(C), subject to the condition L(C)=const, for a closed curve C, or an open arc C. It is clear that the solution is to be g...

We study Lipschitz differentiability spaces, a class of metric measure spaces introduced by Cheeger in [8]. We show that if an Ahlfors regular Lipschitz differentiability space has charts of maximal dimension, then, at almost every point, all its tangents are uniformly rectifiable. In particular, at almost every point, such a space admits a tangent that is isometric to a finite-dimensional Bana...

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...

Let D be a Jordan domain in the complex plane with rectifiable boundary C. Let Aq(D) denote the Bers space with norm || ||,. We prove that if/6 A„(D), 2<q<cc, then there exist functions ■s„W = 2£=i l/(z — zn!c), znltEC for k = \,■ ■ ,n, such that \Un— f\\«—"-0This result does not hold for 1 <q^2 even when D is a disc.

In this note we describe the notion of a rectifiable scan and consider some applications [DH1], [DH2] to Plateau-type minimization problems. “Scans” were first introduced in the work [HR1] of Tristan Rivière and the second author to adequately describe certain bubbling phenomena. There, the behavior of certain W 1,3 weakly convergent sequences of smooth maps from 4 dimensional domains into S le...

J. J. Schâffer [3] introduced an interesting parameter for normed linear spaces. It is termed girth and is the infimum of the lengths of all centrally symmetric simple closed rectifiable curves which lie in the boundary of the unit ball. More precisely, let X be a Banach space with norm denoted by || -|| and with dim X^2. K curve in X will be a rectifiable geometric curve defined by Busemann [l...