Uniform homeomorphisms between spheres induced by interpolation methods

Abstract

M. Daher [Canad. Math. Bull. 38 (1995), pp. 286–294] showed that if ( X 0 , X 1 ) (X_0, X_1) is a regular couple of uniformly convex spaces then the unit spheres of the complex interpolation spaces X θ X_{\theta } and X η X_{\eta } are uniformly homeomorphic for every 0 > θ , η > 1 0 > \theta , \eta > 1 . We show that this is a rather general phenomenon of the interpolation methods described by the discrete framework of interpolation of Lindemulder and Lorist [A discrete framework for the interpolation of Banach spaces, https://arxiv.org/abs/2105.08373, 2021].