Related papers: Dilatation structures with the Radon-Nikodym prope…
We introduce length dilatation structures on metric spaces, tempered dilatation structures and coherent projections and explore the relations between these objects and the Radon-Nikodym property and Gamma-convergence of length functionals.…
We study dentable maps from a closed convex subset of a Banach space into a metric space as an attempt of generalize the Radon-Nikod\'ym property to a "less linear" frame. We note that a certain part of the theory can be developed in rather…
A dilatation structure is a concept in between a group and a differential structure. In this article we study fundamental properties of dilatation structures on metric spaces. This is a part of a series of papers which show that such a…
We prove that every Banach space, not necessarily separable, can be isometrically embedded into a $\mathcal L_{\infty}$-space in a way that the corresponding quotient has the Radon-Nikodym and the Schur properties. As a consequence, we…
Based on the notion of dilatation structure arXiv:math/0608536, we give an intrinsic treatment to sub-riemannian geometry, started in the paper arXiv:0706.3644 . Here we prove that regular sub-riemannian manifolds admit dilatation…
We prove the differentiability of Lipschitz maps X-->V, where X is a complete metric measure space satisfying a doubling condition and a Poincar\'e inequality, and V is a Banach space with the Radon Nikodym Property (RNP). The proof depends…
It is well-known that the class of piecewise smooth curves together with a smooth Riemannian metric induces a metric space structure on a manifold. However, little is known about the minimal regularity needed to analyze curves and…
We demonstrate the necessity of a Poincar\'e type inequality for those metric measure spaces that satisfy Cheeger's generalization of Rademacher's theorem for all Lipschitz functions taking values in a Banach space with the Radon-Nikodym…
We provide an axiomatic approach to the theory of local tangent cones of regular sub-Riemannian manifolds and the differentiability of mappings between such spaces. This axiomatic approach relies on a notion of a dilation structure which is…
It is well known that every bounded below and non increasing sequence in the real line converges. We give a version of this result valid in Banach spaces with the Radon-Nikodym property, thus extending a former result of A. Proch\'azka.
We obtain a result in the spirit of the well-known W. Schachermeyer and H. P. Rosenthal research about the equivalence between Radon-Nikodym and Krein-Milman properties, by showing that, for closed, bounded and convex subsets C of a…
We show that any dimension $6$ nearly K\"ahler (or nearly para-K\"ahler) geometry arises as a projective manifold equipped with a $\textrm{G}_2^{(*)}$ holonomy reduction. In the converse direction we show that if a projective manifold is…
Banach spaces without the Radon-Nikod\'ym property are characterized as spaces containing bilipschitz images of thick families of geodesics defined as follows. A family $T$ of geodesics joining points $u$ and $v$ in a metric space is called…
A dilatation structure on a metric space, arXiv:math/0608536v4, is a notion in between a group and a differential structure, accounting for the approximate self-similarity of the metric space. The basic objects of a dilatation structure are…
We introduce and study equivariant versions of the Radon-Nikod\'ym property for Banach spaces, together with the closely related notions such as dentability, the Bishop-Phelps and Krein-Milman properties, and Lindenstrauss' property A, all…
In this paper we continue the study of dilatation structures, introduced in math.MG/0608536 . A dilatation structure on a metric space is a kind of enhanced self-similarity. By way of examples this is explained here with the help of the…
In this paper we study two properties viz. property-$U$ and property-$SU$ of a subspace $Y$ of a Banach space which correspond to the uniqueness of the Hahn-Banach extension of each linear functional in $Y^*$ and in addition to that this…
We prove the equivalence of two seemingly very different ways of generalising Rademacher's theorem to metric measure spaces. One such generalisation is based upon the notion of forming partial derivatives along a very rich structure of…
Some necessary and sufficient conditions are found for Banach function lattices to have the Radon-Nikod\'ym property. Consequently it is shown that an Orlicz space $L_\varphi$ over a non-atomic $\sigma$-finite measure space $(\Omega,…
We prove a Rademacher-type theorem for Lipschitz mappings from a subset of a Carnot group to a Banach homogeneous group, equipped with a suitably weakened Radon-Nikodym property. We provide a metric area formula that applies to these…