scholarly journals Lipschitz extensions on generalized Grushin spaces

2005 ◽  
Vol 53 (1) ◽  
pp. 3-31 ◽  
Author(s):  
Thomas Bieske
Keyword(s):  
Lipschitz Extensions ◽  
Grushin Spaces

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.

Brain and Surface Warping via Minimizing Lipschitz Extensions (PREPRINT)

2006 ◽  
Author(s):  
Facundo Memoli ◽  
Guillermo Sapiro ◽  
Paul Thompson
Keyword(s):  
Lipschitz Extensions

Assouad-Nagata dimension via Lipschitz extensions

2009 ◽  
Vol 171 (1) ◽  
pp. 405-423 ◽  
Author(s):  
N. Brodskiy ◽  
J. Dydak ◽  
J. Higes ◽  
A. Mitra
Keyword(s):  
Lipschitz Extensions ◽  
Nagata Dimension

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.

Simultaneous Lipschitz extensions

2005 ◽  
Vol 60 (6) ◽  
pp. 1057-1076
Author(s):  
A Yu Brudnyi ◽  
Yu A Brudnyĭ
Keyword(s):  
Lipschitz Extensions
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