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arXiv · 1411.3915

On the $(\beta)$-distortion of some infinite graphs

Florent Pierre Baudier
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Abstract

We show a distortion lower bound of $\Omega(\log(h)^{1/p})$ when embedding the countably branching hyperbolic tree of height $h$ into a Banach space with an equivalent norm satisfying Rolewicz property $(\beta)$ with modulus of power type $p>1$. Similarly we show that a distortion lower bound of $\Omega(l^{1/p})$ is incurred when embedding the parasol graphs with $l$ levels into a Banach space with the above property. We discuss the optimality of our results as well as several applications.

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Florent Pierre Baudier. 2014-11-14. On the $(\beta)$-distortion of some infinite graphs. https://arxiv.org/abs/1411.3915

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