Editing Convex set (section)

== Definitions ==
[[File:Convex supergraph.svg|right|thumb|A [[convex function|function]] is convex if and only if its [[Epigraph (mathematics)|epigraph]], the region (in green) above its [[graph of a function|graph]] (in blue), is a convex set.]]
Let {{mvar|S}} be a [[vector space]] or an [[affine space]] over the [[real number]]s, or, more generally, over some [[ordered field]] (this includes Euclidean spaces, which are affine spaces). A [[subset]] {{mvar|C}} of {{mvar|S}} is '''convex''' if, for all {{mvar|x}} and {{mvar|y}} in {{mvar|C}}, the [[line segment]] connecting {{mvar|x}} and {{mvar|y}} is included in {{mvar|C}}. 

This means that the [[affine combination]] {{math|(1 − ''t'')''x'' + ''ty''}} belongs to {{mvar|C}} for all {{mvar|x,y}} in {{mvar|C}} and {{mvar|t}} in the [[interval (mathematics)|interval]] {{math|[0, 1]}}. This implies that convexity is invariant under [[affine transformation]]s. Further, it implies that a convex set in a [[real number|real]] or [[complex number|complex]] [[topological vector space]] is [[path-connected]] (and therefore also [[connected space|connected]]).

A set {{mvar|C}} is '''{{visible anchor|strictly convex}}''' if every point on the line segment connecting {{mvar|x}} and {{mvar|y}} other than the endpoints is inside the [[Interior (topology)|topological interior]] of {{mvar|C}}. A closed convex subset is strictly convex if and only if every one of its [[Boundary (topology)|boundary points]] is an [[extreme point]].<ref>{{Halmos A Hilbert Space Problem Book 1982|p=5}}</ref>

A set {{mvar|C}} is '''[[absolutely convex]]''' if it is convex and [[balanced set|balanced]].

===Examples===
The convex [[subset]]s of {{math|'''R'''}} (the set of real numbers) are the intervals and the points of {{math|'''R'''}}. Some examples of convex subsets of the [[Euclidean plane]] are solid [[regular polygon]]s, solid triangles, and intersections of solid triangles. Some examples of convex subsets of a [[Euclidean space|Euclidean 3-dimensional space]] are the [[Archimedean solid]]s and the [[Platonic solid]]s. The [[Kepler-Poinsot polyhedra]] are examples of non-convex sets.

=== Non-convex set ===
A set that is not convex is called a ''non-convex set''. A [[polygon]] that is not a [[convex polygon]] is sometimes called a [[concave polygon]],<ref>{{cite book |first=Jeffrey J. |last=McConnell |year=2006 |title=Computer Graphics: Theory Into Practice |isbn=0-7637-2250-2 |page=[https://archive.org/details/computergraphics0000mcco/page/130 130] |publisher=Jones & Bartlett Learning |url=https://archive.org/details/computergraphics0000mcco/page/130 }}.</ref> and some sources more generally use the term ''concave set'' to mean a non-convex set,<ref>{{MathWorld|title=Concave|id=Concave}}</ref> but most authorities prohibit this usage.<ref>{{cite book|title=Analytical Methods in Economics|first=Akira|last=Takayama|publisher=University of Michigan Press|year=1994|isbn=9780472081356|url=https://books.google.com/books?id=_WmZA0MPlmEC&pg=PA54|page=54|quote=An often seen confusion is a "concave set". Concave and convex functions designate certain classes of functions, not of sets, whereas a convex set designates a certain class of sets, and not a class of functions. A "concave set" confuses sets with functions.}}</ref><ref>{{cite book|title=An Introduction to Mathematical Analysis for Economic Theory and Econometrics|first1=Dean|last1=Corbae|first2=Maxwell B.|last2=Stinchcombe|first3= Juraj|last3=Zeman|publisher=Princeton University Press|year=2009|isbn=9781400833085|url=https://books.google.com/books?id=j5P83LtzVO8C&pg=PT347|page=347|quote=There is no such thing as a concave set.}}</ref>

The [[Complement (set theory)|complement]] of a convex set, such as the [[epigraph (mathematics)|epigraph]] of a [[concave function]], is sometimes called a ''reverse convex set'', especially in the context of [[mathematical optimization]].<ref>{{cite journal | last = Meyer | first = Robert | journal = SIAM Journal on Control and Optimization | mr = 0312915 | pages = 41–54 | title = The validity of a family of optimization methods | volume = 8 | year = 1970| doi = 10.1137/0308003 | url = https://minds.wisconsin.edu/bitstream/handle/1793/57508/TR28.pdf?sequence=1 }}.</ref>