Some results on metric trees
Using isometric embedding of metric trees into Banach spaces, this paper will investigate barycenters, type and cotype, and various measures of compactness of metric trees. A metric tree (T, d) is a metric space such that between any two of its points there is a unique arc that is isometric to an interval in ℝ. We begin our investigation by examining isometric embeddings of metric trees into Banach spaces. We then investigate the possible images x₀ = π((x₁ + ... + xₙ)/n), where π is a contractive retraction from the ambient Banach space X onto T (such a π always exists) in order to understand the "metric" barycenter of a family of points x₁,...,xₙ in a tree T. Further, we consider the metric properties of trees such as their type and cotype. We identify various measures of compactness of metric trees (their covering numbers, ϵ-entropy and Kolmogorov widths) and the connections between them. Additionally, we prove that the limit of the sequence of Kolmogorov widths of a metric tree is equal to its ball measure of non-compactness.
- Department of Mathematics, Claremont McKenna College, Claremont, CA 91711, U.S.A.
- Department of Mathematics, University of California-Irvine, Irvine, CA 92697, U.S.A.
- Department of Mathematics,, University of Illinois at Urbana-Champaign, Urbana, IL 61801, U.S.A.