# Wydawnictwa / Banach Center Publications / Wszystkie tomy

## Some results on metric trees

### Tom 91 / 2010

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

#### Streszczenie

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.

#### Autorzy

• Asuman Güven AksoyDepartment of Mathematics,
Claremont McKenna College
Claremont, CA 91711, U.S.A.
e-mail
• Timur OikhbergDepartment of Mathematics
University of California-Irvine
Irvine, CA 92697, U.S.A.
and
Department of Mathematics,
University of Illinois at Urbana-Champaign
Urbana, IL 61801, U.S.A.
e-mail

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