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Hyperconvexity of \(\mathbb{R}\)-trees. (English) Zbl 0913.54030

An \(\mathbb{R}\)-tree is a nonempty metric space \(M\) satisfying: (a) Any two points \(p,q\in M\) are joined by a unique metric segment \([p,q]\); (b) If \(p,q,r\in M\), then \([p,q]\cap[p,r]=[p,w]\) for some \(w\in M\); (c) If \(p,q,r\in M\) and \([p,q]\cap[q,r]=\{q\}\), then \([p,q]\cup[q,r]=[p,r]\). \(\mathbb{R}\)-trees were introduced by J. Tits in [Contrib. to Algebra, Collect. Pap. dedic. E. Kolchin, 377-388 (1977; Zbl 0373.20039)]. A metric space \((M,d)\) is hyperconvex if \(\bigcap_\alpha B(x_\alpha; r_\alpha) \neq \emptyset\) for any collection \(\{B(x_\alpha; r_\alpha)\}\) of closed balls in \(M\) for which \(d(x_\alpha,x_\beta)\leq r_\alpha+r_\beta\). The main results is the following:
Theorem. A metric space \(M\) is a complete \(\mathbb{R}\)-tree if and only if \(M\) is hyperconvex and has unique metric segments. The author also observes that any complete \(\mathbb{R}\)-tree can be viewed as a nonexpansive retract of a Banach space, and that this suggests a new approach to the study of fixed point theory in \(\mathbb{R}\)-trees.

MSC:

54H12 Topological lattices, etc. (topological aspects)
05C12 Distance in graphs
54H25 Fixed-point and coincidence theorems (topological aspects)
51K05 General theory of distance geometry

Citations:

Zbl 0373.20039
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