Inter-rater reliability - Revision history

173.227.247.250: /* Krippendorff’s Alpha */

2014-12-02T18:41:10Z

Krippendorff’s Alpha

← Older revision		Revision as of 20:41, 2 December 2014
Line 1:		Line 1:
	~~Hi there, I am~~ ~~extended auto warranty Yoshiko~~ [http://En.~~wikipedia.org~~/~~wiki~~/~~Warranty Villareal~~] ~~but I by no means truly liked that name.~~ [http://Www.~~toyotacertified~~.com/~~warranty~~.~~html Managing people~~] ~~is how I make money and it's something I really enjoy~~. ~~What she enjoys doing is bottle tops collecting~~ and ~~she auto warranty~~ is ~~attempting~~ to ~~make~~ it ~~a occupation~~. ~~Her family extended [http://www.shipzula.com/blogs/post/26656 extended car warranty] [http://mcb-law.net/great-ideas-about-auto-repair-that-you-can-use/ car warranty] members lives in Delaware~~ but ~~she requirements to transfer because of her family members~~.<br><br>My ~~web site~~ - [http://~~Postarticle~~.~~net~~/~~30665~~/~~automotive~~-~~basics~~-~~the~~-~~way~~-to-~~employ~~-a-~~professional~~-~~mechanic/ extended auto warranty~~]		The individual who wrote the article is called Jayson Hirano and he completely digs that title. Invoicing is online [http://chungmuroresidence.com/xe/reservation_branch2/152663 best psychic readings] chat ([http://Www.Aseandate.com/index.php?m=member_profile&p=profile&id=13352970 Www.Aseandate.com]) my profession. The favorite pastime for him and his kids is to perform lacross and he'll be starting some thing else along with it. North Carolina is the place he enjoys most but now he is contemplating other options.<br><br>My website - [http://www.zavodpm.ru/blogs/glennmusserrvji/14565-great-hobby-advice-assist-allow-you-get-going online reader]

129.118.195.128: /* Sources of inter-rater disagreement */ New paragraph distinguishes multi-rater situations from those without a need for multiple raters, and provides examples of each.

2014-03-03T15:32:38Z

Sources of inter-rater disagreement: New paragraph distinguishes multi-rater situations from those without a need for multiple raters, and provides examples of each.

← Older revision		Revision as of 17:32, 3 March 2014
Line 1:		Line 1:
	~~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~~.		Hi there, I am extended auto warranty Yoshiko [http://En.wikipedia.org/wiki/Warranty Villareal] but I by no means truly liked that name. [http://Www.toyotacertified.com/warranty.html Managing people] is how I make money and it's something I really enjoy. What she enjoys doing is bottle tops collecting and she auto warranty is attempting to make it a occupation. Her family extended [http://www.shipzula.com/blogs/post/26656 extended car warranty] [http://mcb-law.net/great-ideas-about-auto-repair-that-you-can-use/ car warranty] members lives in Delaware but she requirements to transfer because of her family members.<br><br>My web site - [http://Postarticle.net/30665/automotive-basics-the-way-to-employ-a-professional-mechanic/ extended auto warranty]

	~~== Hyperconvexity ==~~

	~~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,~~
	~~#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~~
	~~#if ''F'' is any family of closed balls~~
	~~::<math>{\bar B}_r(p) = \{q \mid d(p,q) \le r\}</math>~~
	:~~such that each pair of balls in ''F'' meet, then there exists a point ''x'' common to all the balls in ''F''.~~

	~~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>''.

	~~== Injectivity ==~~

	~~A [[retract (metric geometry)\|retraction]] of a metric space ''X'' is a function ''&fnof;'' mapping ''X'' to a subspace of itself, such that~~
	~~# for all ''x'', ''&fnof;''(''&fnof;''(''x'')) = ''&fnof;''(''x''); that is, ''&fnof;'' is the [[identity function]] on its image, and~~
	~~# for all ''x'' and ''y'', ''d''(''&fnof;''(''x''), ''&fnof;''(''y'')) ≤ ''d''(''x'', ''y''); that is, ''&fnof;'' is [[nonexpansive mapping\|nonexpansive]]~~.
	~~A ''retract'' of a space ''X'' is a subspace of ''X'' that is an image of a retraction.~~
	~~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''.~~

	~~== Examples ==~~

	~~Examples of hyperconvex metric spaces include~~
	* The real line
	* Any vector space '''R'''<sup>''d''</sup> with the [[Lp space\|L<sub>∞</sub> distance]]
	* [[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~~
	* The [[tight span]] of a metric space
	* Any [[real tree]]
	* Aim(''X'') – see [[Metric space aimed at its subspace]]
	~~Due to the equivalence between hyperconvexity and injectivity, these spaces are all also injective~~.

	~~== Properties ==~~

	~~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]]~~.

	~~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).~~

	~~== References ==~~
	*{{cite journal
	~~\| author1-link = Nachman Aronszajn \| last1 = Aronszajn \| first1 = N. \| last2 = Panitchpakdi \| first2 = P.~~
	~~\| title = Extensions of uniformly continuous transformations and hyperconvex metric spaces~~
	~~\| mr = 0084762~~
	~~\| journal =~~ [~~[Pacific Journal of Mathematics]]~~
	~~\| volume = 6~~
	~~\| year = 1956~~
	~~\| pages = 405–439~~
	~~\| url =~~ http://~~projecteuclid~~.~~org~~/~~Dienst~~/UI/~~1.0~~/~~Summarize~~/~~euclid~~.~~pjm~~/~~1103043960}} Correction (1957), ''Pacific J. Math.'' '''7''': 1729, {{MR\|0092146}}~~.
	*{{cite journal
	~~\| last = Chepoi \| first = Victor~~
	~~\| title = A ''T~~<~~sub~~>X<~~/sub~~>~~'' approach to some results on cuts and metrics~~
	~~\| mr = 1479014~~
	~~\| journal =~~ [~~[Advances in Applied Mathematics]]~~
	~~\| volume = 19~~
	~~\| issue = 4~~
	~~\| year = 1997~~
	~~\| pages = 453–470~~
	~~\| doi = 10.1006/aama.1997.0549}}~~
	*{{cite conference
	~~\| last1 = Espínola \| first1 = R. \| last2 = Khamsi \| first2 = M. A.~~
	~~\| title = Introduction to hyperconvex spaces~~
	~~\| mr = 1904284~~
	~~\| booktitle = Handbook of Metric Fixed Point Theory~~
	~~\| editor = Kirk, W. A.; Sims, B. (Eds.)~~
	~~\| publisher = Kluwer Academic Publishers~~
	~~\| location = Dordrecht~~
	~~\| year = 2001~~
	~~\| url =~~ http://~~drkhamsi~~.~~com~~/~~publication~~/Es-~~Kh.pdf}}~~
	*{{cite journal
	~~\| last = Isbell \| first = J. R. \| authorlink = John R. Isbell~~
	~~\| title = Six theorems about injective metric spaces~~
	~~\| journal = [[Commentarii Mathematici Helvetici]]~~
	~~\| mr = 0182949~~
	~~\| volume = 39~~
	~~\| year = 1964~~
	~~\| pages = 65–76~~
	~~\| doi = 10.1007/BF02566944}}~~
	*{{cite journal
	~~\| last = Sine \| first = R. C.~~
	~~\| title = On nonlinear contraction semigroups in sup norm spaces~~
	~~\| mr = 0548959~~
	~~\| journal = Nonlinear Analysis~~
	~~\| volume = 3~~
	~~\| year = 1979~~
	~~\| pages = 885–890~~
	~~\| doi = 10.1016/0362~~-~~546X(79)90055~~-5
	~~\| issue = 6}}~~
	*{{cite journal
	~~\| last = Soardi \| first = P.~~
	~~\| title = Existence of fixed points of nonexpansive mappings in certain Banach lattices~~
	~~\| mr = 0512051~~
	~~\| journal = [[Proceedings of the American Mathematical Society]]~~
	~~\| volume = 73~~
	~~\| year = 1979~~
	~~\| pages = 25–29~~
	~~\| doi = 10.2307~~/~~2042874~~
	~~\| issue = 1~~
	~~\| jstor = 2042874}}~~

	~~[[Category:Metric geometry]~~]

71.126.104.139: added link to reference

2013-11-11T05:44:31Z

added link to reference

← Older revision		Revision as of 07:44, 11 November 2013
Line 1:		Line 1:
	Hi there. ~~My title~~ is ~~Sophia Meagher although it~~ is ~~not~~ the ~~title~~ on ~~my birth certification~~. ~~To play lacross~~ is ~~something he would by no means give up~~. ~~My working day job~~ is a ~~travel agent~~. ~~Some time~~ in the ~~past she chose~~ to ~~reside~~ in ~~Alaska~~ and ~~her parents live nearby~~.<br><br>~~Feel free~~ to ~~surf~~ to ~~my weblog~~ ... ~~best psychic~~ ([http://si.~~dgmensa.org~~/xe/~~index~~.~~php?document_srl~~=~~48014&mid~~=~~c0102 si~~.~~dgmensa~~.~~org~~])		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.

			== Hyperconvexity ==

			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,
			#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
			#if ''F'' is any family of closed balls
			::<math>{\bar B}_r(p) = \{q \mid d(p,q) \le r\}</math>
			:such that each pair of balls in ''F'' meet, then there exists a point ''x'' common to all the balls in ''F''.

			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>''.

			== Injectivity ==

			A [[retract (metric geometry)\|retraction]] of a metric space ''X'' is a function ''&fnof;'' mapping ''X'' to a subspace of itself, such that
			# for all ''x'', ''&fnof;''(''&fnof;''(''x'')) = ''&fnof;''(''x''); that is, ''&fnof;'' is the [[identity function]] on its image, and
			# for all ''x'' and ''y'', ''d''(''&fnof;''(''x''), ''&fnof;''(''y'')) ≤ ''d''(''x'', ''y''); that is, ''&fnof;'' is [[nonexpansive mapping\|nonexpansive]].
			A ''retract'' of a space ''X'' is a subspace of ''X'' that is an image of a retraction.
			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''.

			== Examples ==

			Examples of hyperconvex metric spaces include
			* The real line
			* Any vector space '''R'''<sup>''d''</sup> with the [[Lp space\|L<sub>∞</sub> distance]]
			* [[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
			* The [[tight span]] of a metric space
			* Any [[real tree]]
			* Aim(''X'') – see [[Metric space aimed at its subspace]]
			Due to the equivalence between hyperconvexity and injectivity, these spaces are all also injective.

			== Properties ==

			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]].

			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).

			== References ==
			*{{cite journal
			\| author1-link = Nachman Aronszajn \| last1 = Aronszajn \| first1 = N. \| last2 = Panitchpakdi \| first2 = P.
			\| title = Extensions of uniformly continuous transformations and hyperconvex metric spaces
			\| mr = 0084762
			\| journal = [[Pacific Journal of Mathematics]]
			\| volume = 6
			\| year = 1956
			\| pages = 405–439
			\| url = http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.pjm/1103043960}} Correction (1957), ''Pacific J. Math.'' '''7''': 1729, {{MR\|0092146}}.
			*{{cite journal
			\| last = Chepoi \| first = Victor
			\| title = A ''T<sub>X</sub>'' approach to some results on cuts and metrics
			\| mr = 1479014
			\| journal = [[Advances in Applied Mathematics]]
			\| volume = 19
			\| issue = 4
			\| year = 1997
			\| pages = 453–470
			\| doi = 10.1006/aama.1997.0549}}
			*{{cite conference
			\| last1 = Espínola \| first1 = R. \| last2 = Khamsi \| first2 = M. A.
			\| title = Introduction to hyperconvex spaces
			\| mr = 1904284
			\| booktitle = Handbook of Metric Fixed Point Theory
			\| editor = Kirk, W. A.; Sims, B. (Eds.)
			\| publisher = Kluwer Academic Publishers
			\| location = Dordrecht
			\| year = 2001
			\| url = http://drkhamsi.com/publication/Es-Kh.pdf}}
			*{{cite journal
			\| last = Isbell \| first = J. R. \| authorlink = John R. Isbell
			\| title = Six theorems about injective metric spaces
			\| journal = [[Commentarii Mathematici Helvetici]]
			\| mr = 0182949
			\| volume = 39
			\| year = 1964
			\| pages = 65–76
			\| doi = 10.1007/BF02566944}}
			*{{cite journal
			\| last = Sine \| first = R. C.
			\| title = On nonlinear contraction semigroups in sup norm spaces
			\| mr = 0548959
			\| journal = Nonlinear Analysis
			\| volume = 3
			\| year = 1979
			\| pages = 885–890
			\| doi = 10.1016/0362-546X(79)90055-5
			\| issue = 6}}
			*{{cite journal
			\| last = Soardi \| first = P.
			\| title = Existence of fixed points of nonexpansive mappings in certain Banach lattices
			\| mr = 0512051
			\| journal = [[Proceedings of the American Mathematical Society]]
			\| volume = 73
			\| year = 1979
			\| pages = 25–29
			\| doi = 10.2307/2042874
			\| issue = 1
			\| jstor = 2042874}}

			[[Category:Metric geometry]]

69.140.226.125: /* Further reading */

2012-07-04T22:12:37Z

← Older revision		Revision as of 17:32, 3 March 2014
Line 1:		Line 1:
	~~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~~.		Hi there, I am extended auto warranty Yoshiko [http://En.wikipedia.org/wiki/Warranty Villareal] but I by no means truly liked that name. [http://Www.toyotacertified.com/warranty.html Managing people] is how I make money and it's something I really enjoy. What she enjoys doing is bottle tops collecting and she auto warranty is attempting to make it a occupation. Her family extended [http://www.shipzula.com/blogs/post/26656 extended car warranty] [http://mcb-law.net/great-ideas-about-auto-repair-that-you-can-use/ car warranty] members lives in Delaware but she requirements to transfer because of her family members.<br><br>My web site - [http://Postarticle.net/30665/automotive-basics-the-way-to-employ-a-professional-mechanic/ extended auto warranty]

	~~== Hyperconvexity ==~~

	~~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,~~
	~~#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~~
	~~#if ''F'' is any family of closed balls~~
	~~::<math>{\bar B}_r(p) = \{q \mid d(p,q) \le r\}</math>~~
	:~~such that each pair of balls in ''F'' meet, then there exists a point ''x'' common to all the balls in ''F''.~~

	~~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>''.

	~~== Injectivity ==~~

	~~A [[retract (metric geometry)\|retraction]] of a metric space ''X'' is a function ''&fnof;'' mapping ''X'' to a subspace of itself, such that~~
	~~# for all ''x'', ''&fnof;''(''&fnof;''(''x'')) = ''&fnof;''(''x''); that is, ''&fnof;'' is the [[identity function]] on its image, and~~
	~~# for all ''x'' and ''y'', ''d''(''&fnof;''(''x''), ''&fnof;''(''y'')) ≤ ''d''(''x'', ''y''); that is, ''&fnof;'' is [[nonexpansive mapping\|nonexpansive]]~~.
	~~A ''retract'' of a space ''X'' is a subspace of ''X'' that is an image of a retraction.~~
	~~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''.~~

	~~== Examples ==~~

	~~Examples of hyperconvex metric spaces include~~
	* The real line
	* Any vector space '''R'''<sup>''d''</sup> with the [[Lp space\|L<sub>∞</sub> distance]]
	* [[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~~
	* The [[tight span]] of a metric space
	* Any [[real tree]]
	* Aim(''X'') – see [[Metric space aimed at its subspace]]
	~~Due to the equivalence between hyperconvexity and injectivity, these spaces are all also injective~~.

	~~== Properties ==~~

	~~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]]~~.

	~~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).~~

	~~== References ==~~
	*{{cite journal
	~~\| author1-link = Nachman Aronszajn \| last1 = Aronszajn \| first1 = N. \| last2 = Panitchpakdi \| first2 = P.~~
	~~\| title = Extensions of uniformly continuous transformations and hyperconvex metric spaces~~
	~~\| mr = 0084762~~
	~~\| journal =~~ [~~[Pacific Journal of Mathematics]]~~
	~~\| volume = 6~~
	~~\| year = 1956~~
	~~\| pages = 405–439~~
	~~\| url =~~ http://~~projecteuclid~~.~~org~~/~~Dienst~~/UI/~~1.0~~/~~Summarize~~/~~euclid~~.~~pjm~~/~~1103043960}} Correction (1957), ''Pacific J. Math.'' '''7''': 1729, {{MR\|0092146}}~~.
	*{{cite journal
	~~\| last = Chepoi \| first = Victor~~
	~~\| title = A ''T~~<~~sub~~>X<~~/sub~~>~~'' approach to some results on cuts and metrics~~
	~~\| mr = 1479014~~
	~~\| journal =~~ [~~[Advances in Applied Mathematics]]~~
	~~\| volume = 19~~
	~~\| issue = 4~~
	~~\| year = 1997~~
	~~\| pages = 453–470~~
	~~\| doi = 10.1006/aama.1997.0549}}~~
	*{{cite conference
	~~\| last1 = Espínola \| first1 = R. \| last2 = Khamsi \| first2 = M. A.~~
	~~\| title = Introduction to hyperconvex spaces~~
	~~\| mr = 1904284~~
	~~\| booktitle = Handbook of Metric Fixed Point Theory~~
	~~\| editor = Kirk, W. A.; Sims, B. (Eds.)~~
	~~\| publisher = Kluwer Academic Publishers~~
	~~\| location = Dordrecht~~
	~~\| year = 2001~~
	~~\| url =~~ http://~~drkhamsi~~.~~com~~/~~publication~~/Es-~~Kh.pdf}}~~
	*{{cite journal
	~~\| last = Isbell \| first = J. R. \| authorlink = John R. Isbell~~
	~~\| title = Six theorems about injective metric spaces~~
	~~\| journal = [[Commentarii Mathematici Helvetici]]~~
	~~\| mr = 0182949~~
	~~\| volume = 39~~
	~~\| year = 1964~~
	~~\| pages = 65–76~~
	~~\| doi = 10.1007/BF02566944}}~~
	*{{cite journal
	~~\| last = Sine \| first = R. C.~~
	~~\| title = On nonlinear contraction semigroups in sup norm spaces~~
	~~\| mr = 0548959~~
	~~\| journal = Nonlinear Analysis~~
	~~\| volume = 3~~
	~~\| year = 1979~~
	~~\| pages = 885–890~~
	~~\| doi = 10.1016/0362~~-~~546X(79)90055~~-5
	~~\| issue = 6}}~~
	*{{cite journal
	~~\| last = Soardi \| first = P.~~
	~~\| title = Existence of fixed points of nonexpansive mappings in certain Banach lattices~~
	~~\| mr = 0512051~~
	~~\| journal = [[Proceedings of the American Mathematical Society]]~~
	~~\| volume = 73~~
	~~\| year = 1979~~
	~~\| pages = 25–29~~
	~~\| doi = 10.2307~~/~~2042874~~
	~~\| issue = 1~~
	~~\| jstor = 2042874}}~~

	~~[[Category:Metric geometry]~~]