Roundness and metric type

We prove that if X is a Banach space containing l(p)(n) uniformly in n, and if Y is a metric space with metric type q > p, then the inverse of any uni form homeomorphism T from X onto Y cannot satisfy a Lipschitz condition for large distances of order alpha < q/p. It follows that if Y is a midpoint-c onvex subset of a Banach space Z with type q larger than the type supremum of a Banach space X, then X and Y cannot be uniformly homeomorphic. In part icular, we prove the non-existence of uniform homeomorphisms between certai n non-commutative L-p-spaces and midpoint-convex subsets of another such sp ace. We also prove that if a Banach space X has cotype infimum q larger tha n two, then it has maximal generalized roundness zero and maximal roundness at most q'. As a consequence, infinite-dimensional C*-algebras are seen to have maximal generalized roundness zero and maximal roundness one. (C) 200 0 Academic Press.