en>LokiClock: /* An equivalence relation defined by LFTs */ name mystery group

2014-01-17T22:27:39Z

An equivalence relation defined by LFTs: name mystery group

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	~~The individual who wrote~~ the ~~article~~ is ~~known as Jayson Hirano~~ and ~~he completely digs~~ that ~~title~~. ~~Invoicing~~ is ~~what I do~~ for a ~~living but I~~'~~ve always needed my own company~~. ~~My spouse~~ and ~~I reside~~ in ~~Mississippi but now I'm considering~~ other ~~options~~. ~~What me and my family members adore~~ is ~~bungee leaping but I've been using~~ on ~~new issues lately~~.<br><br>~~Feel~~ [~~http~~://~~www~~.~~edmposts~~.~~com~~/~~build-~~a~~-beautiful-organic-garden-using-these-ideas~~/ ~~free psychic readings~~] ~~to visit my weblog accurate~~ [~~http:~~//~~isaworld~~.pe.~~kr/?document_srl=392088 live psychic reading] predictions~~ ([~~http:~~//~~fashionlinked~~.~~com/index~~.~~php?do~~=~~/profile~~-~~13453/info/ click through the up coming web site~~])		[[File:Convex metric illustration2.png\|right\|thumb\|An illustration of a convex metric space.]]
			In [[mathematics]], '''convex metric spaces''' are, intuitively, [[metric space]]s with the property any "segment" joining two points in that space has other points in it besides the endpoints.

			Formally, consider a [[metric space]] (''X'', ''d'') and let ''x'' and ''y'' be two points in ''X''. A point ''z'' in ''X'' is said to be ''between'' ''x'' and ''y'' if all three points are distinct, and

			: <math>d(x, z)+d(z, y)=d(x, y),\,</math>

			that is, the [[triangle inequality]] becomes an equality. A '''convex metric space''' is a metric space (''X'', ''d'') such that, for any two distinct points ''x'' and ''y'' in ''X'', there exists a third point ''z'' in ''X'' lying between ''x'' and ''y''.

			Metric convexity:
			* does not imply convexity in the usual sense for subsets of [[Euclidean space]] (see the example of the rational numbers)
			* nor does it imply [[path-connectedness]] (see the example of the rational numbers)
			* nor does it imply [[geodesic convexity]] for [[Riemannian manifolds]] (consider, for example, the Euclidean plane with a closed disc removed).

			==Examples==
			* Euclidean spaces, that is, the usual three-dimensional space and its analogues for other dimensions, are convex metric spaces. Given any two distinct points <math>x</math> and <math>y</math> in such a space, the set of all points <math>z</math> satisfying the above "triangle equality" forms the [[line segment]] between <math>x</math> and <math>y,</math> which always has other points except <math>x</math> and <math>y,</math> in fact, it has a [[Continuum (set theory)\|continuum]] of points.
			[[File:Circle as convex metric space.png\|right\|thumb\|A circle as a convex metric space.]]
			* Any [[convex set]] in a Euclidean space is a convex metric space with the induced Euclidean norm. For [[closed set]]s the [[Contraposition\|converse]] is also true: if a closed subset of a Euclidean space together with the induced distance is a convex metric space, then it is a convex set (this is a particular case of a more general statement to be discussed below).
			* A [[circle]] is a convex metric space, if the distance between two points is defined as the length of the shortest arc on the circle connecting them.

			==Metric segments==
			Let <math>(X, d)</math> be a metric space (which is not necessarily convex). A subset <math>S</math> of <math>X</math> is called a '''metric segment''' between two distinct points <math>x</math> and <math>y</math> in <math>X,</math> if there exists a closed interval <math>[a, b]</math> on the real line and an [[isometry]]

			: <math>\gamma:[a, b] \to X,\,</math>

			such that <math>\gamma([a, b])=S,</math> <math>\gamma(a)=x</math> and <math>\gamma(b)=y.</math>

			It is clear that any point in such a metric segment <math>S</math> except for the "endpoints" <math>x</math> and <math>y</math> is between <math>x</math> and <math>y.</math> As such, if a metric space <math>(X, d)</math> admits metric segments between any two distinct points in the space, then it is a convex metric space.

			The [[Contraposition\|converse]] is not true, in general. The [[rational number]]s form a convex metric space with the usual distance, yet there exists no segment connecting two rational numbers which is made up of rational numbers only. If however, <math>(X, d)</math> is a convex metric space, and, in addition, it is [[Complete metric space\|complete]], one can prove that for any two points <math>x\ne y</math> in <math>X</math> there exists a metric segment connecting them (which is not necessarily unique).

			==Convex metric spaces and convex sets==

			As mentioned in the examples section, closed subsets of Euclidean spaces are convex metric spaces if and only if they are convex sets. It is then natural to think of convex metric spaces as generalizing the notion of convexity beyond Euclidean spaces, with usual linear segments replaced by metric segments.

			It is important to note, however, that metric convexity defined this way does not have one of the most important properties of Euclidean convex sets, that being that the intersection of two convex sets is convex. Indeed, as mentioned in the examples section, a circle, with the distance between two points measured along the shortest arc connecting them, is a ([[Complete metric space\|complete]]) convex metric space. Yet, if <math>x</math> and <math>y</math> are two points on a circle diametrally opposite to each other, there exist two metric segments connecting them (the two arcs into which these points split the circle), and those two arcs are metrically convex, but their intersection is the set <math>\{x, y\}</math> which is not metrically convex.

			==See also==
			*[[Intrinsic metric]]
			==References==

			*{{cite book
			\| last = Khamsi
			\| first = Mohamed A.
			\| coauthors = [[William Arthur Kirk\|Kirk, William A.]]
			\| title = An Introduction to Metric Spaces and Fixed Point Theory
			\| publisher = Wiley-IEEE
			\| date = 2001
			\| pages =
			\| isbn = 0-471-41825-0
			}}

			*{{cite book
			\| last = Kaplansky
			\| first = Irving
			\| title = Set Theory and Metric Spaces
			\| publisher = American Mathematical Society
			\| date = 2001
			\| pages =
			\| isbn = 0-8218-2694-8
			}}

			[[Category:Metric geometry]]
			[[Category:Convex geometry]]

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The individual who wrote the article is known as Jayson Hirano and he completely digs that title. Invoicing is what I do for a living but I've always needed my own company. My spouse and I reside in Mississippi but now I'm considering other options. What me and my family members adore is bungee leaping but I've been using on new issues lately.<br><br>Feel [http://www.edmposts.com/build-a-beautiful-organic-garden-using-these-ideas/ free psychic readings] to visit my weblog accurate [http://isaworld.pe.kr/?document_srl=392088 live psychic reading] predictions ([http://fashionlinked.com/index.php?do=/profile-13453/info/ click through the up coming web site])

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