Extending Lipschitz maps into C(K)-spaces
- Cite this article as:
- Kalton, N.J. Isr. J. Math. (2007) 162: 275. doi:10.1007/s11856-007-0099-2
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We show that if K is a compact metric space then C(K) is a 2-absolute Lipschitz retract. We then study the best Lipschitz extension constants for maps into C(K) from a given metric space M, extending recent results of Lancien and Randrianantoanina. They showed that a finite-dimensional normed space which is polyhedral has the isometric extension property for C(K)-spaces; here we show that the same result holds for spaces with Gateaux smooth norm or of dimension two; a three-dimensional counterexample is also given. We also show that X is polyhedral if and only if every subset E of X has the universal isometric extension property for C(K)-spaces. We also answer a question of Naor on the extension of Hölder continuous maps.