Generalized Roundness and Negative Type

摘要

In this paper we exhibit the equivalence of Enflo's nonlinear notion of generalized roundness and the classical embedding notion of negative type. This enables us to develop a rudimentary theory of generalized roundness and to give applications to the L_p-spaces. In particular, we show that for p > 2and n ≥ 3, the n-dimensional l_p spaces fail to have generalized roundness q for all q > 0. The notions of roundness and generalized roundness were introduced by Enflo in [E1], [E2], and [E3] to study the uniform structure of metric spaces. We begin by recalling some material from these papers. However, we make some slight alterations to Enflo's original definitions to allow easier exposition later.