Sphere equivalence, Banach expanders, and extrapolation

We study the Banach spectral gap lambda_1(G;X,p) of finite graphs G for pairs (X,p) of Banach spaces and exponents. We define the notion of sphere equivalence between Banach spaces and show a generalization of Matousek's extrapolation for Banach spaces sphere equivalent to uniformly convex ones. As a byproduct, we prove that expanders are automatically expanders with respects to (X,p) for any X sphere equivalent to a uniformly curved Banach space and for any p strictly bigger than 1.

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