Related papers: Extension, separation and isomorphic reverse isope…

A multiple-valued function $f:X\to {\bf Q}_Q(Y)$ is essentially a rule assigning $Q$ unordered and non necessarily distinct elements of $Y$ to each element of $X$. We study the Lipschitz extension problem in this context by using two…

If $X$ is a metric space, then its finite subset spaces $X(n)$ form a nested sequence under natural isometric embeddings $X = X(1)\subset X(2) \subset \cdots$. It was previously established, by Kovalev when $X$ is a Hilbert space and, by…

Wenger and Young proved that the pair $(\mathbb{R}^m,\mathbb{H}^n)$ has the Lipschitz extension property for $m \leq n$ where $\mathbb{H}^n$ is the sub-Riemannian Heisenberg group. That is, for some $C>0$, any $L$-Lipschitz map from a…

We consider two questions on the geometry of Lipschitz free $p$-spaces $\mathcal F_p$, where $0<p\leq 1$, over subsets of finite-dimensional vector spaces. We solve an open problem and show that if $(\mathcal M, \rho)$ is an infinite…

If a metric subspace $M^{o}$ of an arbitrary metric space $M$ carries a doubling measure $\mu$, then there is a simultaneous linear extension of all Lipschitz functions on $M^{o}$ ranged in a Banach space to those on $M$. Moreover, the norm…

It is hereby established that the set of Lipschitz functions $f:\mathcal{U}\rightarrow \mathbb{R}$ ($\mathcal{U}$ nonempty open subset of $\ell_{d}^{1}$) with maximal Clarke subdifferential contains a linear subspace of uncountable…

In this paper, we provide an infinite metric space $M$ such that the set $\mbox{SNA}(M)$ of strongly norm-attaining Lipschitz functions does not contain a subspace which is isometric to $c_0$. This answers a question posed by Antonio…

We show that no matter what subset of a normed space is given, a typical 1-Lipschitz mapping into a Banach space is non-differentiable at a typical point of the set in a very strong sense: the derivative ratio approximates, on arbitrary…

Given a Banach space $X$ and a real number $\alpha\ge 1$, we write: (1) $D(X)\le\alpha$ if, for any locally finite metric space $A$, all finite subsets of which admit bilipschitz embeddings into $X$ with distortions $\le C$, the space $A$…

We consider Lipschitz maps with values in quasi-metric spaces and extend such maps to finitely many points. We prove that in this context every 1-Lipschitz map admits an extension such that its Lipschitz constant is bounded from above by…

Let $X$ be a normed space of a finite dimension at least two, and $C\subsetneq X$ a closed convex set with nonempty interior. We are interested in extending Lipschitz quasiconvex functions on $C$ to quasiconvex functions on $X$. We show…

We obtain an optimal deviation from the mean upper bound \begin{equation} D(x)\=\sup_{f\in \F}\mu\{f-\E_{\mu} f\geq x\},\qquad\ \text{for}\ x\in\R\label{abstr} \end{equation} where $\F$ is the class of the integrable, Lipschitz functions on…

We prove that for certain subsets $M \subseteq \mathbb{R}^N$, $N \geqslant 1$, the Lipschitz-free space $\mathcal{F}(M)$ has the metric approximation property (MAP), with respect to any norm on $\mathbb{R}^N$. In particular,…

We find general conditions under which Lipschitz-free spaces over metric spaces are isomorphic to their infinite direct $\ell_1$-sum and exhibit several applications. As examples of such applications we have that Lipschitz-free spaces over…

For a domain $\Omega$ in a finite-dimensional space $E$, we consider the space $M=(\Omega,d)$ where $d$ is the intrinsic distance in $\Omega$. We obtain an isometric representation of the space $\mathrm{Lip}_{0}(M)$ as a subspace of…

In the present paper we introduce and study the Lipschitz retractional structure of metric spaces. This topic was motivated by the analogous projectional structure of Banach spaces, a topic that has been thoroughly investigated. The more…

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 establish the sharp rate of continuity of extensions of $\mathbb{R}^m$-valued $1$-Lipschitz maps from a subset $A$ of $\mathbb{R}^n$ to a $1$-Lipschitz maps on $\mathbb{R}^n$. We consider several cases when there exists a $1$-Lipschitz…

We conjecture that whenever $M$ is a metric space of density at most continuum, then the space of Lipschitz functions is $w^*$-separable. We prove the conjecture for several classes of metric spaces including all the Banach spaces with a…

We investigate the first-order differential calculus over extended metric-topological measure spaces. The latter are quartets $\mathbb X=(X,\tau,{\sf d},\mathfrak m)$, given by an extended metric space $(X,{\sf d})$ together with a weaker…