Volume 20, issue 3 (2016)

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The metric geometry of the Hamming cube and applications

Florent Baudier, Daniel Freeman, Thomas Schlumprecht and András Zsák

Geometry & Topology 20 (2016) 1427–1444
Abstract

The Lipschitz geometry of segments of the infinite Hamming cube is studied. Tight estimates on the distortion necessary to embed the segments into spaces of continuous functions on countable compact metric spaces are given. As an application, the first nontrivial lower bounds on the $C\left(K\right)$–distortion of important classes of separable Banach spaces, where $K$ is a countable compact space in the family $\left\{\left[0,\omega \right],\left[0,\omega \cdot 2\right],\dots ,\left[0,{\omega }^{2}\right],\dots ,\left[0,{\omega }^{k}\cdot n\right],\dots ,\left[0,{\omega }^{\omega }\right]\right\}$ are obtained.

 In memory of Luis Sánchez-González

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Keywords
countable compact metric space, bi-Lipschitz embedding, $C(K)$ space
Mathematical Subject Classification 2010
Primary: 46B20, 46B80
Secondary: 46B85