On the (β)-distortion of some infinite graphs

الأصل · EN

We show a distortion lower bound of Ω((h)¹/ᵖ) when embedding the countably branching hyperbolic tree of height h into a Banach space with an equivalent norm satisfying Rolewicz property (β) with modulus of power type p>1. Similarly we show that a distortion lower bound of Ω(l¹/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.