Related papers: On bilipschitz extensions in real Banach spaces

Given a superreflexive Banach space $X$, and a set $E \subset X$, we characterise the $1$-jets $(f,G)$ on $E$ that admit $C^{1,\omega}$ convex extensions $(F,DF)$ to all of $X$; where $\omega$ is any admissible modulus of continuity…

Let $X$ be a real Banach space. A subset $B$ of the dual unit sphere of $X$ is said to be a boundary for $X$, if every element of $X$ attains its norm on some functional in $B$. The well-known Boundary Problem originally posed by Godefroy…

We show that any non-zero Banach space with a separable dual contains a totally disconnected, closed and bounded subset S of Hausdorff dimension 1 such that every Lipschitz function on the space is Fr\'echet differentiable somewhere in S.

In this paper we show that every sequence (F_n) of finite dimensional subspaces of a real or complex Banach space with increasing dimensions can be ``refined'' to yield an F.D.D. (G_n), still having increasing dimensions, so that either…

For a real normed space $X$, we study the $n$-dual space of $\left(X,\left\Vert \cdot \right\Vert \right) $ and show that the space is a Banach space. Meanwhile, for a real normed space $X$ of dimension $d\geq n$ which satisfies property…

We prove a commutative Gelfand--Naimark type theorem, by showing that the set $C_s(X)$ of continuous bounded (real or complex valued) functions with separable support on a locally separable metrizable space $X$ (provided with the supremum…

M. Daher gave conditions so that the spheres of the spaces in the interior of a complex interpolation scale are uniformly homeomorphic. We look for sufficient conditions for the validity of this result and related ones on the extremes of…

In this paper, we investigate holomorphic mappings $F$ on the unit ball $\mathbb{B}$ of a complex Banach space of the form $F(x)=f(x)x$, where $f$ is a holomorphic function on $\mathbb{B}$. First, we investigate criteria for univalence,…

Let $E$ be an arbitrary subset of a Banach space $X$, $f: E \rightarrow \mathbb{R}$ be a function, and $G:E \rightrightarrows X^*$ be a set-valued mapping. We give necessary and sufficient conditions on $f, G$ for the existence of a…

We prove that every bi-Lipschitz embedding of the unit circle into the plane can be extended to a bi-Lipschitz map of the unit disk with linear bounds on the constants involved. This answers a question raised by Daneri and Pratelli.…

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…

We prove that if X is an infinite dimensional Banach lattice with a weak unit then there exists a probability space (Omega, Sigma,mu) so that the unit sphere S(L_1(Omega, Sigma, mu) is uniformly homeomorphic to the unit sphere S(X) if and…

Theorems on continuous extension on boundary for one class of open discrete mappings between Riemannian manifolds are obtained. In particular, there is proved that, open discrete ring $Q$-mappings $f:D\rightarrow D^{\,\prime}$ are extend to…

Let $X$ and $Y$ be Banach or normed linear spaces and $F\subset X$ a closed set. We apply our recent extension theorem for vector-valued Baire one functions arXiv:1512.03717 to obtain an extension theorem for vector-valued functions…

Let \(X\) be a compact metric space and \(E\) be a Banach space. \(\lip (X, E)\) denotes the Banach space of all \(E\)-valued little Lipschitz functions on \(X\). We show that \(\lip (X, E)^{**}\) is isometrically isomorphic to Banach space…

Let $X$ be a closed semialgebraic set of dimension $k.$ If $n\ge 2k+1$, then there is a bi-Lipschitz and semialgebraic embedding of $X$ into $\Bbb R^n.$ Moreover, if $n \ge 2k+2$, then this embedding is unique (up to a bi-Lipschitz and…

We study the isometric extension problem for H\"{o}lder maps from subsets of any Banach space into $c_0$ or into a space of continuous functions. For a Banach space $X$, we prove that any $\alpha$-H\"{o}lder map, with $0<\alpha\leq 1$, from…

We find conditions for a smooth nonlinear map $f:U\rightarrow V$ between open subsets of Hilbert or Banach spaces to be locally convex in the sense that for some $c$ and each positive $\varepsilon<c$ the image $% f(B_\varepsilon(x))$ of…

We give an alternative proof of the stable manifold theorem as an application of the (right and left) inverse mapping theorem on a space of sequences. We investigate the diffeomorphism class of the global stable manifold, a problem which in…

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…