Some Results on Metric Trees

TL;DR

This paper explores the geometric and metric properties of trees by embedding them into Banach spaces, analyzing barycenters, type, cotype, and measures of compactness, revealing new connections between these concepts.

Contribution

It introduces new methods to analyze metric trees via isometric embeddings into Banach spaces and investigates their barycenters, type, cotype, and compactness measures.

Findings

Existence of contractive retractions onto metric trees

Characterization of metric barycenters in Banach spaces

Equivalence of Kolmogorov widths limit and ball measure of non-compactness

Abstract

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.