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Barriers in metric spaces. (English) Zbl 1207.54028

Call a subset of a connected topological space a barrier if it is connected and its complement is disconnected. The authors investigate certain barriers in the tight span \(T(D)\) of a metric space.
Let \(X\) be a set and let \(D\) be a metric for \(X\). Write
\[ T(D)= \{f\in\mathbb{R}^X: f(x)= \sup(D(x,y)- f(y)), y\in X,\forall x\in X\}; \] \(T(D)\) is called the tight span of \(D\); the authors consider sets of the form \(T_{(f,\varepsilon)}(D)= T(D)- B_\varepsilon(f)\), where \(B_\varepsilon(f)\) is the closed unit ball in \(T(D)\), with \(\varepsilon> 0\).
Consider: \(\Gamma=\) the graph with vertex-set \(\text{supp}_\varepsilon(f)\) and edge set
\[ \left\{(a,b)\in \begin{pmatrix}\text{supp}_\varepsilon(f)\\ 2\end{pmatrix}: f(a)+ f(b)> ab+\varepsilon\right\}. \]
\(\pi_0(T_{f,\varepsilon}(D))=\) the set of connected components of \(T_{(f,\varepsilon)}(D)\).
\(\pi_0(\Gamma)\) the set of connected components of \(\Gamma\). Then the authors establish:
Result 1. There exists a canonical surjective mapping
\[ \pi_f: \pi_0(T_{(f,\varepsilon)}(D))\to \pi_0(\Gamma). \]
Result 2. Given a bipartition of \(\text{supp}_\varepsilon(f)\) into two nonempty subsets \(A\) and \(B\) such that the corresponding open sets \(O_{(f,\varepsilon)}(A)\) and \(O_{(f,\varepsilon)}(B)\) of \(T_{(f,\varepsilon)}(D)\) form a bipartition of \(T_{(f,\varepsilon)}(D)\) with \(f(a)+ f(b)\leq ab+ 2\varepsilon\), \(\forall a\in A\) and \(\forall b\in B\), then \(B_\varepsilon(f)\) is a barrier in \(T(D)\).

MSC:

54D05 Connected and locally connected spaces (general aspects)
54E35 Metric spaces, metrizability
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References:

[1] Dress, A., Trees, tight extensions of metric spaces, and the cohomological dimension of certain groups: A note on combinatorial properties of metric spaces, Adv. Math., 53, 321-402 (1984) · Zbl 0562.54041
[2] Dress, A., The tight span of metric spaces, (Phylogenetic Combinatorics (2008), Shaker Publishing Company: Shaker Publishing Company Greifswald, Germany), pp. 111-181
[3] Dress, A.; Moulton, V.; Terhalle, W., T-Theory: An overview, European J. Combin., 17, 161-175 (1996) · Zbl 0853.54027
[4] Dress, A.; Huber, K.; Koolen, J.; Moulton, V., An algorithm for computing virtual cut points in finite metric spaces, (COCOA 2007. COCOA 2007, Lecture Notes in Computer Science, vol. 4616 (2007)), 4-10 · Zbl 1175.05049
[5] Dress, A.; Huber, K.; Koolen, J.; Moulton, V., Compatible decompositions and block realizations of finite metric spaces, European J. Combin., 29, 1617-1633 (2008) · Zbl 1179.05029
[6] Dress, A.; Huber, K.; Koolen, J.; Moulton, V., Cut points in metric spaces, Appl. Math. Lett., 21, 545-548 (2008) · Zbl 1142.54007
[7] A. Dress, V. Moulton, T. Wu, A topological approach to tree (re-)construction (submitted for publication); A. Dress, V. Moulton, T. Wu, A topological approach to tree (re-)construction (submitted for publication)
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