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| In [[metric geometry]], an '''injective metric space''', or equivalently a '''hyperconvex metric space''', is a [[metric space]] with certain properties generalizing those of the real line and of [[Chebyshev distance|L<sub>∞</sub> distances]] in higher-dimensional [[vector space]]s. These properties can be defined in two seemingly different ways: hyperconvexity involves the intersection properties of closed balls in the space, while injectivity involves the [[isometry|isometric embeddings]] of the space into larger spaces. However it is a theorem of Aronszajn and Panitchpakdi (1956; see e.g. Chepoi 1997) that these two different types of definitions are equivalent.
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| == Hyperconvexity ==
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| A metric space ''X'' is said to be '''hyperconvex''' if it is [[convex metric|convex]] and its closed [[Ball (mathematics)|balls]] have the binary [[Helly family|Helly property]]. That is,
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| #any two points ''x'' and ''y'' can be connected by the [[isometry|isometric image]] of a line segment of length equal to the distance between the points (i.e. ''X'' is a path space), and
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| #if ''F'' is any family of closed balls
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| ::<math>{\bar B}_r(p) = \{q \mid d(p,q) \le r\}</math>
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| :such that each pair of balls in ''F'' meet, then there exists a point ''x'' common to all the balls in ''F''. | |
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| Equivalently, if a set of points ''p<sub>i</sub>'' and radii ''r<sub>i</sub> > 0'' satisfies ''r<sub>i</sub>'' + ''r<sub>j</sub>'' ≥ ''d''(''p<sub>i</sub>'',''p<sub>j</sub>'') for each ''i'' and ''j'', then there is a point ''q'' of the metric space that is within distance ''r<sub>i</sub>'' of each ''p<sub>i</sub>''.
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| == Injectivity ==
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| A [[retract (metric geometry)|retraction]] of a metric space ''X'' is a function ''ƒ'' mapping ''X'' to a subspace of itself, such that
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| # for all ''x'', ''ƒ''(''ƒ''(''x'')) = ''ƒ''(''x''); that is, ''ƒ'' is the [[identity function]] on its image, and
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| # for all ''x'' and ''y'', ''d''(''ƒ''(''x''), ''ƒ''(''y'')) ≤ ''d''(''x'', ''y''); that is, ''ƒ'' is [[nonexpansive mapping|nonexpansive]].
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| A ''retract'' of a space ''X'' is a subspace of ''X'' that is an image of a retraction.
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| A metric space ''X'' is said to be '''injective''' if, whenever ''X'' is [[isometry|isometric]] to a subspace ''Z'' of a space ''Y'', that subspace ''Z'' is a retract of ''Y''.
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| == Examples ==
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| Examples of hyperconvex metric spaces include
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| * The real line
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| * Any vector space '''R'''<sup>''d''</sup> with the [[Lp space|L<sub>∞</sub> distance]]
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| * [[taxicab geometry|Manhattan distance]] (''L''<sub>1</sub>) in the plane (which is equivalent up to rotation and scaling to the ''L''<sub>∞</sub>), but not in higher dimensions
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| * The [[tight span]] of a metric space
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| * Any [[real tree]]
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| * Aim(''X'') – see [[Metric space aimed at its subspace]]
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| Due to the equivalence between hyperconvexity and injectivity, these spaces are all also injective.
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| == Properties ==
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| In an injective space, the radius of the [[circumradius|minimum ball]] that contains any set ''S'' is equal to half the [[diameter]] of ''S''. This follows since the balls of radius half the diameter, centered at the points of ''S'', intersect pairwise and therefore by hyperconvexity have a common intersection; a ball of radius half the diameter centered at a point of this common intersection contains all of ''S''. Thus, injective spaces satisfy a particularly strong form of [[Jung's theorem]].
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| Every injective space is a [[complete space]] (Aronszajn and Panitchpakdi 1956), and every [[metric map]] (or, equivalently, [[short map|nonexpansive mapping, or short map]]) on a bounded injective space has a [[Fixed-point theorem|fixed point]] (Sine 1979; Soardi 1979). A metric space is injective if and only if it is an [[injective object]] in the [[category (mathematics)|category]] of [[category of metric spaces|metric spaces and metric maps]]. For additional properties of injective spaces see Espínola and Khamsi (2001).
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| == References ==
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| *{{cite journal
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| | author1-link = Nachman Aronszajn | last1 = Aronszajn | first1 = N. | last2 = Panitchpakdi | first2 = P.
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| | title = Extensions of uniformly continuous transformations and hyperconvex metric spaces
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| | mr = 0084762 | |
| | journal = [[Pacific Journal of Mathematics]]
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| | volume = 6
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| | year = 1956
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| | pages = 405–439
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| | url = http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.pjm/1103043960}} Correction (1957), ''Pacific J. Math.'' '''7''': 1729, {{MR|0092146}}.
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| *{{cite journal
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| | last = Chepoi | first = Victor
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| | title = A ''T<sub>X</sub>'' approach to some results on cuts and metrics
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| | mr = 1479014
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| | journal = [[Advances in Applied Mathematics]]
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| | volume = 19
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| | issue = 4
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| | year = 1997
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| | pages = 453–470
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| | doi = 10.1006/aama.1997.0549}}
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| *{{cite conference
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| | last1 = Espínola | first1 = R. | last2 = Khamsi | first2 = M. A.
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| | title = Introduction to hyperconvex spaces
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| | mr = 1904284
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| | booktitle = Handbook of Metric Fixed Point Theory
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| | editor = Kirk, W. A.; Sims, B. (Eds.)
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| | publisher = Kluwer Academic Publishers
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| | location = Dordrecht
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| | year = 2001
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| | url = http://drkhamsi.com/publication/Es-Kh.pdf}}
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| *{{cite journal
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| | last = Isbell | first = J. R. | authorlink = John R. Isbell
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| | title = Six theorems about injective metric spaces
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| | journal = [[Commentarii Mathematici Helvetici]]
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| | mr = 0182949
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| | volume = 39
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| | year = 1964
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| | pages = 65–76
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| | doi = 10.1007/BF02566944}}
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| *{{cite journal
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| | last = Sine | first = R. C.
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| | title = On nonlinear contraction semigroups in sup norm spaces
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| | mr = 0548959
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| | journal = Nonlinear Analysis
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| | volume = 3
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| | year = 1979
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| | pages = 885–890
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| | doi = 10.1016/0362-546X(79)90055-5
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| | issue = 6}}
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| *{{cite journal
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| | last = Soardi | first = P.
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| | title = Existence of fixed points of nonexpansive mappings in certain Banach lattices
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| | mr = 0512051
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| | journal = [[Proceedings of the American Mathematical Society]]
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| | volume = 73
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| | year = 1979
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| | pages = 25–29
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| | doi = 10.2307/2042874
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| | issue = 1
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| | jstor = 2042874}}
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| [[Category:Metric geometry]]
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