Some results on metric trees

Volume 91 / 2010

Asuman Güven Aksoy, Timur Oikhberg Banach Center Publications 91 (2010), 9-34 MSC: Primary 54E35; Secondary 54E45, 54E50, 05C05, 47H09, 51F99. DOI: 10.4064/bc91-0-1


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 $\mathbb{R}$. We begin our investigation by examining isometric embeddings of metric trees into Banach spaces. We then investigate the possible images $x_0=\pi ((x_1+\ldots+x_n)/n)$, where $\pi$ is a contractive retraction from the ambient Banach space $X$ onto $T$ (such a $\pi$ always exists) in order to understand the “metric” barycenter of a family of points $ x_1, \ldots ,x_n$ 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, $\epsilon$-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.


  • Asuman Güven AksoyDepartment of Mathematics,
    Claremont McKenna College
    Claremont, CA 91711, U.S.A.
  • Timur OikhbergDepartment 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.

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