Some results on metric trees

Asuman Güven Aksoy, and Timur Oikhberg. "Some results on metric trees." Banach Center Publications 91.1 (2010): 9-34. <http://eudml.org/doc/286660>.

@article{AsumanGüvenAksoy2010,
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 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.},
author = {Asuman Güven Aksoy, Timur Oikhberg},
journal = {Banach Center Publications},
keywords = {metric tree; barycenter; metric type; metric cotype; measure of non-compactness; Kolmogorov widths; -entropy},
language = {eng},
number = {1},
pages = {9-34},
title = {Some results on metric trees},
url = {http://eudml.org/doc/286660},
volume = {91},
year = {2010},
}

TY - JOUR
AU - Asuman Güven Aksoy
AU - Timur Oikhberg
TI - Some results on metric trees
JO - Banach Center Publications
PY - 2010
VL - 91
IS - 1
SP - 9
EP - 34
AB - 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.
LA - eng
KW - metric tree; barycenter; metric type; metric cotype; measure of non-compactness; Kolmogorov widths; -entropy
UR - http://eudml.org/doc/286660
ER -