Related papers: On Fr\'echet differentiability of Lipschitz maps b…

We study the relationships between Gateaux, weak Hadamard and Frechet differentiability and their bornologies for Lipschitz and for convex functions. In particular, Frechet and weak Hadamard differentiabily coincide for all Lipschitz…

We consider convex series of molecules in Lipschitz-free spaces, i.e. elements of the form $\mu=\sum_n \lambda_n \frac{\delta_{x_n}-\delta_{y_n}}{d(x_n,y_n)}$ such that $\|\mu\|=\sum_n |\lambda_n |$. We characterise these elements in terms…

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…

the main goal of this paper is to prove that any Banach space X, that every dual ball in X** is weak* -separable, or every weak* -closed convex subset in X** is weak* -separable, or every norm-closed convex set in X* is constructible,…

We prove that several classical Banach space properties are equivalent to separability for the class of Lipschitz-free spaces, including Corson's property ($\mathcal{C}$), Talponen's Countable Separation Property, or being a G\^ateaux…

We show the existence of Lipschitz-free spaces verifying the Point of Continuity Property with arbitrarily high weak-fragmentability index. For this purpose, we use a generalized construction of the countably branching diamond graphs. As a…

Let (X, d) be a quasi-convex, complete and separable metric space with reference probability measure m. We prove that the set of of real valued Lipschitz function with non zero point-wise Lipschitz constant m-almost everywhere is residual,…

Given a category of objects, it is both useful and important to know if all the objects in the category may be realised as sub-objects -- via morphisms in the given category -- of a single object in that category enjoying some nice…

We show that if n>1 then there exists a Lebesgue null set in R^n containing a point of differentiability of each Lipschitz function mapping from R^n to R^(n-1); in combination with the work of others, this completes the investigation of…

It is known that a Lipschitz continuous map from the Euclidean domain to a metric space is metrically differentiable almost everywhere. When the metric space is a Banach space dual to separable, the metric differential has its linear…

For a measure space $\Omega$ we extend the theory of Orlicz spaces generated by an even convex integrand $\varphi \colon \Omega \times X \to \left[ 0, \infty \right]$ to the case when the range Banach space $X$ is arbitrary. Besides…

This paper deals with the extension of a classical theorem by R. Phelps on the G\^ateaux differentiability of Lipschitz functions on separable Banach spaces to the non-separable case. The extension of the theorem is not possible for general…

A remarkable theorem of R. C. James is the following: suppose that $X$ is a Banach space and $C \subseteq X$ is a norm bounded, closed and convex set such that every linear functional $x^* \in X^*$ attains its supremum on $C$; then $C$ is a…

Let n>2 and X be a Banach space of dimension strictly greater than n. We show there exists a directionally porous set P in X for which the set of C^1 surfaces of dimension n meeting P in positive measure is not meager. If X is separable…

Rademacher theorem asserts that Lipschitz continuous functions between Euclidean spaces are differentiable almost everywhere. In this work we extend this result to set-valued maps using an adequate notion of set-valued differentiability…

It follows from recent results of V. Bakhtin, R. Oleinik, and the second named author that, given a metric space $\mathcal{X}$, a continuous map $\gamma\colon [a,b] \to \mathcal{X}$ is a map of bounded variation if and only if $f \circ…

We give several structural results concerning the Lipschitz-free spaces $\mathcal F(M)$, where $M$ is a metric space. We show that $\mathcal F(M)$ contains a complemented copy of $\ell_1(\Gamma)$, where $\Gamma=\text{dens}(M)$. If $\mathcal…

For any non-trivial convex and bounded subset $C$ of a Banach space, we show that outside of a $\sigma$-porous subset of the space of non-expansive mappings $C\to C$, all mappings have the maximal Lipschitz constant one witnessed locally at…

We prove a general principle satisfied by weakly precompact sets of Lipschitz-free spaces. By this principle, certain infinite dimensional phenomena in Lipschitz-free spaces over general metric spaces may be reduced to the same phenomena in…

We construct a differentiable locally Lipschitz function $f$ in $\mathbb{R}^{N}$ with the property that for every convex body $K\subset \mathbb{R}^N$ there exists $\bar x \in \mathbb{R}^N$ such that $K$ coincides with the set $\partial_L…