scholarly journals What Happens to a Manifold Under a Bi-Lipschitz Map?

2016 ◽  
Vol 57 (3) ◽  
pp. 641-673 ◽  
Author(s):  
Armin Eftekhari ◽  
Michael B. Wakin
Keyword(s):  
Lipschitz Map

scholarly journals A DEFINABLE -ADIC ANALOGUE OF KIRSZBRAUN’S THEOREM ON EXTENSIONS OF LIPSCHITZ MAPS

2015 ◽  
Vol 17 (1) ◽  
pp. 39-57 ◽  
Author(s):  
Raf Cluckers ◽  
Florent Martin
Keyword(s):  
Direct Application ◽  
Lipschitz Maps ◽  
Lipschitz Map ◽  
Lipschitz Extensions ◽  
Topological Closure

A direct application of Zorn’s lemma gives that every Lipschitz map $f:X\subset \mathbb{Q}_{p}^{n}\rightarrow \mathbb{Q}_{p}^{\ell }$ has an extension to a Lipschitz map $\widetilde{f}:\mathbb{Q}_{p}^{n}\rightarrow \mathbb{Q}_{p}^{\ell }$. This is analogous to, but easier than, Kirszbraun’s theorem about the existence of Lipschitz extensions of Lipschitz maps $S\subset \mathbb{R}^{n}\rightarrow \mathbb{R}^{\ell }$. Recently, Fischer and Aschenbrenner obtained a definable version of Kirszbraun’s theorem. In this paper, we prove in the $p$-adic context that $\widetilde{f}$ can be taken definable when $f$ is definable, where definable means semi-algebraic or subanalytic (or some intermediary notion). We proceed by proving the existence of definable Lipschitz retractions of $\mathbb{Q}_{p}^{n}$ to the topological closure of $X$ when $X$ is definable.

scholarly journals Sobolev Extensions of Lipschitz Mappings into Metric Spaces

2017 ◽  
Vol 2019 (8) ◽  
pp. 2241-2265
Author(s):  
Scott Zimmerman
Keyword(s):  
Heisenberg Group ◽  
Bounded Domain ◽  
Compact Subset ◽  
Metric Space ◽  
Metric Spaces ◽  
Extension Property ◽  
Lipschitz Mapping ◽  
Lipschitz Extension ◽  
Lipschitz Mappings ◽  
Lipschitz Map

Abstract 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 subset of $\mathbb{R}^m$ into $\mathbb{H}^n$ can be extended to a $CL$-Lipschitz mapping on $\mathbb{R}^m$. In this article, we construct Sobolev extensions of such Lipschitz mappings with no restriction on the dimension $m$. We prove that any Lipschitz mapping from a compact subset of $\mathbb{R}^m$ into $\mathbb{H}^n$ may be extended to a Sobolev mapping on any bounded domain containing the set. More generally, we prove this result in the case of mappings into any Lipschitz $(n-1)$-connected metric space.

scholarly journals A planar bi-Lipschitz extension theorem

2015 ◽  
Vol 8 (3) ◽  
Author(s):  
Sara Daneri ◽  
Aldo Pratelli
Keyword(s):  
Extension Theorem ◽  
Lipschitz Extension ◽  
Lipschitz Map

AbstractWe prove that, given a planar bi-Lipschitz map

Blowing-Up and Whitney (a)-Regularity

1989 ◽  
Vol 32 (4) ◽  
pp. 482-485 ◽  
Author(s):  
Tzee-Char Kuo ◽  
David J. A. Trotman ◽  
Li Pei Xin
Keyword(s):  
Strong Form ◽  
Lipschitz Map ◽  
Blowing Up

AbstractWhen a pair of adjacent C1 strata (X, Y) is Whitney (a)- regular at a point , there is, up to homeomorphism, just one germ at 0 of intersections S ∩ X where S is a Lipschitz transversal to Y at 0 (S is the graph of a Lipschitz map from The proof uses a blowing-up construction and a strong form of Verdier's (w)-regularity.

scholarly journals Lipschitz Extensions to Finitely Many Points

2018 ◽  
Vol 6 (1) ◽  
pp. 174-191 ◽  
Author(s):  
Giuliano Basso
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Metric Spaces ◽  
Lipschitz Constant ◽  
Target Space ◽  
Square Root ◽  
Lipschitz Maps ◽  
Lipschitz Map ◽  
Lipschitz Extensions

AbstractWe 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 the number of added points plus one. Moreover, we prove that if the source space is a Hilbert space and the target space is a Banach space, then there exists an extension such that its Lipschitz constant is bounded from above by the square root of the total of added points plus one. We discuss applications to metric transforms.

scholarly journals Essential Norm of Composition Operators on Banach Spaces of Hölder Functions

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
A. Jiménez-Vargas ◽  
Miguel Lacruz ◽  
Moisés Villegas-Vallecillos
Keyword(s):  
Dual Space ◽  
Normed Linear Space ◽  
Approximation Property ◽  
Composition Operators ◽  
Essential Norm ◽  
Base Point ◽  
Compact Metric Space ◽  
Hölder Functions ◽  
Finite Dimensional ◽  
Lipschitz Map

Let(X,d)be a pointed compact metric space, let0<α<1, and letφ:X→Xbe a base point preserving Lipschitz map. We prove that the essential norm of the composition operatorCφinduced by the symbolφon the spaceslip0(X,dα)andLip0(X,dα)is given by the formula‖Cφ‖e=limt→0 sup⁡0<d(x, y)<t(d(φ(x),φ(y))α/d(x,y)α)whenever the dual spacelip0(X,dα)∗has the approximation property. This happens in particular whenXis an infinite compact subset of a finite-dimensional normed linear space.

The Hausdorff dimension of small divisors for lower-dimensional KAM-tori

1992 ◽  
Vol 439 (1906) ◽  
pp. 359-371 ◽  
Keyword(s):  
Hausdorff Dimension ◽  
Nonlinear Partial Differential Equations ◽  
Kam Theory ◽  
Fixed Integer ◽  
Lipschitz Map ◽  
Integer Solutions ◽  
Lower Dimensional Tori ◽  
Distinct Integer ◽  
Nearly Integrable Hamiltonian Systems ◽  
Lower Dimensional

Given a bounded domain Ω ⊂ ℝ m and a Lipschitz map Φ : Ω ⟼ ℝ n , we determine the Hausdorff dimension of sets of points ω ∈ Ω for which the inequality | k ·ω — l·Φ(ω)| < Ψ (| k | + | l |) has infinitely many distinct integer solutions ( k, l )∈ℤ m x ℤ n satisfying | l | ⩽ h , where h is a fixed integer. These sets ‘interpolate’ between the cases h = 0 and h = ∞,which occur in the metric theory of Diophantine approximation of independent and dependent quantities, respectively. They arise, for example, in the perturbation theories of lower-dimensional tori in nearly integrable hamiltonian systems (KAM-theory). Among others, it turns out that their Hausdorff dimension is independent of h and n , it only depends on m and the lower order of Ψ at infinity. Part of this result even extends to the case n = ∞ of infinite co-dimension, which is relevant in the KAM-theory of certain nonlinear partial differential equations.

scholarly journals On Conditions for Unrectifiability of a Metric Space

2014 ◽  
Vol 3 (1) ◽  
Author(s):  
Piotr Hajłasz ◽  
Soheil Malekzadeh
Keyword(s):  
Metric Space ◽  
Implicit Function Theorem ◽  
Metric Spaces ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Implicit Function ◽  
Heisenberg Groups ◽  
Lipschitz Map ◽  
Necessary And Sufficient

Abstract We find necessary and sufficient conditions for a Lipschitz map f : E ⊂ ℝk → X into a metric space to satisfy ℋk(f(E)) = 0. An interesting feature of our approach is that despite the fact that we are dealing with arbitrary metric spaces, we employ a variant of the classical implicit function theorem. Applications include pure unrectifiability of the Heisenberg groups.

scholarly journals Isometric embeddings of polyhedra into Euclidean space

2015 ◽  
Vol 07 (04) ◽  
pp. 677-692 ◽  
Author(s):  
Barry Minemyer
Keyword(s):  
Euclidean Space ◽  
Metric Space ◽  
Isometric Embedding ◽  
Piecewise Linear ◽  
Embedding Theorem ◽  
Embedding Theorems ◽  
Dimensional Simplex ◽  
Isometric Embeddings ◽  
Lipschitz Map ◽  
Hyperbolic Polyhedra

In this paper we consider piecewise linear (pl) isometric embeddings of Euclidean polyhedra into Euclidean space. A Euclidean polyhedron is just a metric space [Formula: see text] which admits a triangulation [Formula: see text] such that each n-dimensional simplex of [Formula: see text] is affinely isometric to a simplex in 𝔼n. We prove that any 1-Lipschitz map from an n-dimensional Euclidean polyhedron [Formula: see text] into 𝔼3n is ϵ-close to a pl isometric embedding for any ϵ > 0. If we remove the condition that the map be pl, then any 1-Lipschitz map into 𝔼2n + 1 can be approximated by a (continuous) isometric embedding. These results are extended to isometric embedding theorems of spherical and hyperbolic polyhedra into Euclidean space by the use of the Nash–Kuiper C1 isometric embedding theorem ([9] and [13]).

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