Related papers: On the strongly subdifferentiable points in Lipsch…

Let $({\mathcal M},\rho)$ be a metric space and let $Y$ be a Banach space. Given a positive integer $m$, let $F$ be a set-valued mapping from ${\mathcal M}$ into the family of all compact convex subsets of $Y$ of dimension at most $m$. In…

We prove that the Lipschitz free space over a certain type of discrete metric space has the Radon-Nikod\'ym property. We also show that the Lipschitz free space over a complete, locally compact metric space has the Schur or approximation…

We call a closed subset M of a Banach space X a free basis of X if it contains the null vector and every Lipschitz map from M to a Banach space Y, which preserves the null vectors can be uniquely extended to a bounded linear map from X to…

Suppose that $S_1$ and $S_2$ are nonempty subsets of a complete metric space $(\mathcal{M},d)$ and $\phi,\psi:S_1\to S_2$ are mappings. The aim of this work is to investigate some conditions on $\phi$ and $\psi$ such that the two functions,…

Let $M$ be a complete, connected Riemannian surface and suppose that $\mathcal{S} \subset M$ is a discrete subset. What can we learn about $M$ from the knowledge of all distances in the surface between pairs of points of $\mathcal{S}$? We…

We consider the problem of strong density of smooth maps in the Sobolev space $ W^{s,p}(Q^{m};\mathcal{N}) $, where $ 0 < s < +\infty $, $ 1 \leq p < +\infty $, $ Q^{m} $ is the unit cube in $ \mathbb{R}^{m} $, and $ \mathcal{N} $ is a…

We give a short proof that any non-zero Euclidean space has a compact subset of Hausdorff dimension one that contains a differentiability point of every real-valued Lipschitz function defined on the space.

We introduce the notion of (almost isometric) local retracts in metric space as a natural non-linear version of the concepts of locally complemented and almost isometric ideals from Banach spaces. We prove that given two metric spaces…

Given any $d$-dimensional Lipschitz Riemannian manifold $(M,g)$ with heat kernel $\mathsf{p}$, we establish uniform upper bounds on $\mathsf{p}$ which can always be decoupled in space and time. More precisely, we prove the existence of a…

We prove the result stated in the title; it is equivalent to the existence of a regular point of the sub-Riemannian exponential mapping. We also prove that the metric is analytic on an open everywhere dense subset in the case of a complete…

We introduce convex integrals of molecules in Lipschitz-free spaces $\mathcal{F}(M)$ as a continuous counterpart of convex series considered elsewhere, based on the de Leeuw representation. Using optimal transport theory, we show that these…

A topological space is iso-dense if it has a dense set of isolated points. A topological space is scattered if each of its non-empty subspaces has an isolated point. In $\mathbf{ZF}$, in the absence of the axiom of choice, basic properties…

We give a sharp condition on the lower local Lipschitz constant of a mapping from a metric space supporting a Poincar\'e inequality to a Banach space with the Radon-Nikodym property that guarantees differentiability at almost every point.…

Let $(X, d)$ be a compact metric space and let $\mathcal{M}(X)$ denote the space of all finite signed Borel measures on $X$. Define $I \colon \mathcal{M}(X) \to \R$ by $I(mu) = \int_X \int_X d(x,y) d\mu(x) d\mu(y)$, and set $M(X) = \sup…

We prove that the Lipschitz-free space over a separable ultrametric space has a monotone Schauder basis and is isomorphic to $\ell_1$. This extends results of A. Dalet using an alternative approach.

Let $\mathcal X$ be an infinite locally compact separable metric space with metric $\rho$ and let $f : \mathcal X \longrightarrow \mathcal X$ be a continuous weakly mixing map. Let $\beta = \sup \big\{ \rho(x, y): \{x, y \} \subset \mathcal…

We prove that all extreme points of the unit ball of a Lipschitz-free space over a compact metric space have finite support. Combined with previous results, this completely characterizes extreme points and implies that all of them are also…

We prove that every element of a Lipschitz-free space admits an expression as a convex series of elements with compact support. As a consequence, we conclude that all extreme points of the unit ball of Lipschitz-free spaces are elementary…

We prove that every continuous mapping from a separable infinite-dimensional Hilbert space $X$ into $\mathbb{R}^{m}$ can be uniformly approximated by $C^\infty$ smooth mappings {\em with no critical points}. This kind of result can be…

We characterize preserved extreme points of Lipschitz-free spaces $\mathcal{F}(X)$ in terms of simple geometric conditions on the underlying metric space $(X,d)$. Namely, each preserved extreme point corresponds to a pair of points $p,q$ in…