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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-bc91-0-1,
author = {Asuman G\"uven Aksoy and Timur Oikhberg},
title = {Some results on metric trees},
journal = {Banach Center Publications},
volume = {89},
year = {2010},
pages = {9-34},
zbl = {1209.54015},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc91-0-1}
}
Asuman Güven Aksoy; Timur Oikhberg. Some results on metric trees. Banach Center Publications, Tome 89 (2010) pp. 9-34. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc91-0-1/