Roundness and Metric Type

We prove that if X is a Banach space containing lpn uniformly in n, and if Y is a metric space with metric type q p, then the inverse of any uniform homeomorphism T from X onto Y cannot satisfy a Lipschitz condition for large distances of order q p. It follows that if Y is a midpoint-convex 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 particular, we prove the non-existence of uniform homeomorphisms between certain non-commutative Lp-spaces and midpoint- convex subsets of another such space. We also prove that if a Banach space X has cotype infimum q larger than 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