Rademacher type and Enflo type coincide

Abstract

A nonlinear analogue of the Rademacher type of a Banach space was introduced in classical work of Enflo. The key feature of Enflo type is that its definition uses only the metric structure of the Banach space, while the definition of Rademacher type relies on its linear structure. We prove that Rademacher type and Enflo type coincide, settling a long-standing open problem in Banach space theory. The proof is based on a novel dimension-free analogue of Pisier’s inequality on the discrete cube.

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Authors

Paata Ivanisvili

Department of Mathematics, North Carolina State University, Raleigh, NC and University of California, Irvine, CA

Ramon van Handel

Program in Applied & Computational Mathematics, Princeton University, Princeton, NJ

Alexander Volberg

Department of Mathematics, Michigan State University, East Lansing, MI