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 an 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.