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STUDIA MATHEMATICA  Issue no. 1 / 2008  
Article: 
ON THE LIPSCHITZ EXTENSION CONSTANT FOR A COMPLEXVALUED LIPSCHITZ FUNCTION. Authors: ALEXANDRU ROŞOIU, DRAGOŞ FRĂŢILĂ. 

Abstract: In order to show that the Lipschitz constant for the extension of a complexvalued Lipschitz function cannot generally be 1, one can use the following example (see Lipschitz Algebras, by N. Weaver, World Scientific, Singapore, 1999, p. 18, Example 1.5.7): Let X = {e, p_{1}, p_{2}, p_{3}} be a metric space such that d(p_{i}, p_{j}) = 1, for all distinct and for all and let X_{0} = {p_{1}, p_{2}, p_{3}} be a subset of X. An isometric map of X_{0} into can be extended to X with an increase in the Lipschitz constant of at least this constant being attained for the function that takes e to the circumcenter of the triangle formed by the points f(p_{i}), for all The purpose of this article is to show that we can loosen somewhat the conditions imposed on d, namely we show that considering a metric space X = {e, p_{1}, p_{2}, p_{3}} such that for all distinct the above increase in the Lipschitz constant for the extended Lipschitz function is preserved.  