Metric Transforms of L ₁-Spaces

Abstract

Let (X,d) be a distance space and let F: ℝ₊ → ℝ₊ be a function such that F(0) = 0. We define the distance space (X,F(d)) by setting

$$F{\left( d \right)_{ij}} = F\left( {{d_{ij}}} \right)\;for\;all\;i,j \in X.$$

(9.0.1)

Following Blumenthal [1953], (X,F(d)) is called a metric transform of (X,d). A general question is to find nontrivial functions F which preserve certain properties, such as metricity, L ₁- or L ₂-embeddability, of the original distance space.