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Israel Journal of Mathematics

, Volume 219, Issue 1, pp 115–161 | Cite as

On Lipschitz extension from finite subsets

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Abstract

We prove that for every n ∈ ℕ there exists a metric space (X, d X), an n-point subset SX, a Banach space (Z, \({\left\| \right\|_Z}\)) and a 1-Lipschitz function f: SZ such that the Lipschitz constant of every function F: XZ that extends f is at least a constant multiple of \(\sqrt {\log n} \). This improves a bound of Johnson and Lindenstrauss [JL84]. We also obtain the following quantitative counterpart to a classical extension theorem of Minty [Min70]. For every α ∈ (1/2, 1] and n ∈ ℕ there exists a metric space (X, d X), an n-point subset S ⊆ X and a function f: S → ℓ2 that is α-Hölder with constant 1, yet the α-Hölder constant of any F: X → ℓ2 that extends f satisfies \({\left\| F \right\|_{Lip\left( \alpha \right)}} > {\left( {\log n} \right)^{\frac{{2\alpha - 1}}{{4\alpha }}}} + {\left( {\frac{{\log n}}{{\log \log n}}} \right)^{{\alpha ^2} - \frac{1}{2}}}\). We formulate a conjecture whose positive solution would strengthen Ball’s nonlinear Maurey extension theorem [Bal92], serving as a far-reaching nonlinear version of a theorem of König, Retherford and Tomczak-Jaegermann [KRTJ80]. We explain how this conjecture would imply as special cases answers to longstanding open questions of Johnson and Lindenstrauss [JL84] and Kalton [Kal04].

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© Hebrew University of Jerusalem 2017

Authors and Affiliations

  1. 1.Mathematics DepartmentPrinceton UniversityPrincetonUSA
  2. 2.The Rachel and Selim Benin School of Computer Science and EngineeringThe Hebrew University of JerusalemJerusalemIsrael

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