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m I changed the introduction paragraph, there were way to many qualifiers and other conditional terms and/or jargon that (if we presume that one, as I did, knows nothing about Convex Sets) in any event does nothing to elucidate the subject. The page, however, is well written and I was able to understand the concept by the perfect illustration by the end of the second paragraph. |
Undid revision 1249226293 by Friendlyshadowbank (talk) Thanks for your attempt to make this less jargon-heavy, but your new lead sentence is totally incorrect. It is not true that a convex set is a line. Move some of the jargon to the end of the paragraph. |
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[[File:Convex polygon illustration2.svg|right|thumb|Illustration of a non-convex set. The line segment joining points ''x'' and ''y'' partially extends outside of the set, illustrated in red, and the intersection of the set with the line occurs in two places, illustrated in black.]]
In [[geometry]], a set of points is '''convex''' if it contains every [[line segment]] between two points in the set. Equivalently, a '''convex set''' or a '''convex region''' is a
▲Equivalently, a '''convex set''' or a '''convex region''' is a subset that intersects every [[line (geometry)|line]] into a single [[line segment]] (possibly empty).<ref>{{cite book|last1=Morris|first1=Carla C.|last2=Stark|first2=Robert M.|title=Finite Mathematics: Models and Applications|date=24 August 2015|publisher=John Wiley & Sons|isbn=9781119015383|page=121|url=https://books.google.com/books?id=ZgJyCgAAQBAJ&q=convex+region&pg=PA121|access-date=5 April 2017|language=en}}</ref><ref>{{cite journal|last1=Kjeldsen|first1=Tinne Hoff|title=History of Convexity and Mathematical Programming|journal=Proceedings of the International Congress of Mathematicians|issue=ICM 2010|pages=3233–3257|doi=10.1142/9789814324359_0187|url=http://www.mathunion.org/ICM/ICM2010.4/Main/icm2010.4.3233.3257.pdf|access-date=5 April 2017|url-status=dead|archive-url=https://web.archive.org/web/20170811100026/http://www.mathunion.org/ICM/ICM2010.4/Main/icm2010.4.3233.3257.pdf|archive-date=2017-08-11}}</ref>
For example, a solid [[cube (geometry)|cube]] is a convex set, but anything that is hollow or has an indent, for example, a [[crescent]] shape, is not convex.
Spaces in which convex sets are defined include the [[Euclidean space]]s, the [[affine space]]s over the [[Real number|reals]], and [[hyperbolic space]]s.
The [[boundary (topology)|boundary]] of a convex set in the plane is always a [[convex curve]]. The intersection of all the convex sets that contain a given subset {{mvar|A}} of Euclidean space is called the [[convex hull]] of {{mvar|A}}. It is the smallest convex set containing {{mvar|A}}.
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