Editing Convex set (section)

{{short description|In geometry, set whose intersection with every line is a single line segment}}
[[File:Convex polygon illustration1.svg|right|thumb|Illustration of a convex set shaped like a deformed circle. The line segment joining points ''x'' and ''y'' lies completely within the set, illustrated in green. Since this is true for any potential locations of two points within the set, the set is convex.]]
[[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.<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.

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}}.

A [[convex function]] is a [[real-valued function]] defined on an [[interval (mathematics)|interval]] with the property that its [[epigraph (mathematics)|epigraph]] (the set of points on or above the [[graph of a function|graph]] of the function) is a convex set. [[Convex minimization]] is a subfield of [[mathematical optimization|optimization]] that studies the problem of minimizing convex functions over convex sets. The branch of mathematics devoted to the study of properties of convex sets and convex functions is called [[convex analysis]].

Spaces in which convex sets are defined include the [[Euclidean space]]s, the [[affine space]]s over the [[real number]]s, and certain [[non-Euclidean geometry|non-Euclidean geometries]].