Low distortion embeddings of some metric graphs into Banach spaces

Abstract

We give a simple example of a countable metric graph M such that M Lipschitz embeds with distortion strictly less than 2 into a Banach space X only if X contains an isomorphic copy of l1. Further we show that, for each ordinal α < ω1, the space C([0, ωα]) does not Lipschitz embed into C(K) with distortion strictly less than 2 unless K(α) ≠ 0. Also \(C\left( {\left[ {0,{\omega ^{{\omega ^\alpha }}}} \right]} \right)\) does not Lipschitz embed into a Banach space X with distortion strictly less than 2 unless Sz(X) ≥ ωα+1.