On uniform and coarse rigidity of $L^p([0,1])$

Abstract

. If X is an almost transitive Banach space with amenable isometry group (for example, if X = L p ([0 , 1]) with 1 < p < ∞ ) and X admits a uniformly continuous map X φ −→ E into a Banach space E satisfying inf k x − y k = r (cid:13)(cid:13) φ ( x ) − φ ( y ) (cid:13)(cid:13) > 0 for some r > 0 (that is, φ is almost uncollapsed ), then X admits a simultaneously uniform and coarse embedding into a Banach space V that is finitely representable in L 2 ( E ).