Geodesic
Spectral Calculus and Lipschitz Extension for Barycentric Metric Spaces
Analysis and Geometry in Metric Spaces, Tome 1 (2013) no. 1, pp. 163-199.

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The metric Markov cotype of barycentric metric spaces is computed, yielding the first class of metric spaces that are not Banach spaces for which this bi-Lipschitz invariant is understood. It is shown that this leads to new nonlinear spectral calculus inequalities, as well as a unified framework for Lipschitz extension, including new Lipschitz extension results for CAT (0) targets. An example that elucidates the relation between metric Markov cotype and Rademacher cotype is analyzed, showing that a classical Lipschitz extension theorem of Johnson, Lindenstrauss and Benyamini is asymptotically sharp.
Mots-clés : Markov cotype, Lipschitz extension, CAT (0) metric spaces, nonlinear spectral gaps 
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Mendel, Manor; Naor, Assaf. Spectral Calculus and Lipschitz Extension for Barycentric Metric Spaces. Analysis and Geometry in Metric Spaces, Tome 1 (2013) no. 1, pp. 163-199. http://geodesic.mathdoc.fr/item/AGMS_2013_1_1_a8/