[[File:Convex polygon illustration2.svg|right|thumb|Illustration of a non-convex set. Since the (red) part of the (black and red) line-segment joining the points x and y lies ''outside'' of the (green) set, the set is non-convex.]]
In [[geometry]], a subset of a [[Euclidean space]], or more generally an [[affine space]] over the [[Real number|reals]], is '''convex''' if, given any two points in the subset, itthe subset contains the whole [[line segment]] that joins them. 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.
