Trimming of metric spaces and the tight span (Q1660284)

In this paper the author extends his theory of trimming from finite to arbitrary metric spaces. Trimming transformations are then used to investigate the tight span \(T(X)\) of a metric space \(X.\) The main result splits \(T(X)\) into a union \(\tau\cup \overline{C}\) of two metric subspaces \(\tau\) and \(\overline{C}.\) The space \(\tau\) is the tight span of a certain quotient \(\overline{X_\infty}\) of \(X.\) Furthermore the space \({\overline C}\) is a disjoint union of metric trees which either do not meet \(\tau\) or meet \(\tau\) at their roots lying in \(\overline {X_\infty} \subset \tau.\) Indeed \(\tau\cap \overline{C}\) is the set of the roots of the trees forming \(\overline{C}.\) In this picture the original space \(X\subset T(X)\) consists of the tips of the branches of the trees.