Related papers: Differentiating maps into L^1 and the geometry of …

Let $X, Y$ be complete metric spaces and $E, F$ be Banach spaces. A bijective linear operator from a space of $E$-valued functions on $X$ to a space of $F$-valued functions on $Y$ is said to be biseparating if $f$ and $g$ are disjoint if…

These notes have the intent to introduce the study of the nonlinear aspects of operator space theory. We investigate some results on the nonlinear theory of Banach spaces which remain valid in the noncommutative case. In particular, we show…

We determine when a quasi-isometry between discrete spaces is at bounded distance from a bilipschitz map. From this we prove a geometric version of the Von Neumann conjecture on amenability. We also get some examples in geometric groups…

We prove a version of the implicit function theorem for Lipschitz mappings $f:\mathbb{R}^{n+m}\supset A \to X$ into arbitrary metric spaces. As long as the pull-back of the Hausdorff content $\mathcal{H}_{\infty}^n$ by $f$ has positive…

Consider $v$ a Lipschitz unit vector field on $R^n$ and $K$ its Lipschitz constant. We show that the maps $S_s:S_s(X) = X + sv(X)$ are invertible for $0\leq |s|<1/K$ and define nonsingular point transformations. We use these properties to…

We investigate metric properties of level sets of horizontally differentiable maps defined on the first Heisenberg group $(\Bbb{H}^1,d_{cc})$ equipped with the standard sub-Riemannian structure. In particular, we present an exhaustive…

Given a Lipschitz map $f$ from a cube into a metric space, we find several equivalent conditions for $f$ to have a Lipschitz factorization through a metric tree. As an application we prove a recent conjecture of David and Schul. The…

The paper concerns the investigation of nonconvex and nondifferentiable integral functionals on general Banach spaces, which may not be reflexive and/or separable. Considering two major subdifferentials of variational analysis, we derive…

We extend the recent result of G. Godefroy which concerns the existence of non-norm attaining Lipschitz maps in order to characterize the norm attainment toward vectors for Lipschitz maps in the general setting of underlying space. The main…

In the literature surrounding the theory of Banach spaces, considerable effort has been invested in exploring the conditions on a Banach space X that characterise X as being an inner product space or as a linearly isomorphic copy of a…

For Lipschitz maps between a metric measure space and a metric space, combining the ideas of Kirchheim's metric differentiability and Cheeger's differentiable structures leads to a Rademacher-type theorem for a notion of metric…

Two definitions for the rectfiability of hypersurfaces in Heisenberg groups $\mathbb{H}^n$ have been proposed: one based on $\mathbb{H}$-regular surfaces, and the other on Lipschitz images of subsets of codimension-$1$ vertical subgroups.…

Persistent homology maps a simplicial complex filtered by elements in $\mathbb R$ to finite formal sums of elements of $\mathbb R_{\leq}^{2} = \{ (b,d) \in \mathbb R^2 \cup \{ \infty \} \mid b < d \}$ called (finite) persistence diagrams.…

We propose a unified approach to the study of isometries on algebras of vector-valued Lipschitz maps and those of continuously differentiable maps by means of the notion of admissible quadruples. We describe isometries on function spaces of…

In this note, we aim to establish a number of embeddings between various function spaces that are frequently considered in the theory of Fourier series. More specifically, we give sufficient conditions for the embeddings $\Phi V[h]\subseteq…

A space X is said to be Lipschitz 1-connected if every L-Lipschitz loop in X bounds a O(L)-Lipschitz disk. A Lipschitz 1-connected space admits a quadratic isoperimetric inequality, but it is unknown whether the converse is true. Cornulier…

For a metric space $X$, we study the space $D^{\infty}(X)$ of bounded functions on $X$ whose infinitesimal Lipschitz constant is uniformly bounded. $D^{\infty}(X)$ is compared with the space $\LIP^{\infty}(X)$ of bounded Lipschitz functions…

This paper studies the geometry of bilipschitz maps $f \colon \mathbb{W} \to \mathbb{H}$, where $\mathbb{H}$ is the first Heisenberg group, and $\mathbb{W} \subset \mathbb{H}$ is a vertical subgroup of co-dimension $1$. The images…

The usual theory of negative type (and $p$-negative type) is heavily dependent on an embedding result of Schoenberg, which states that a metric space isometrically embeds in some Hilbert space if and only if it has 2-negative type. A…

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…