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{{short description|In geometry, set thatwhose intersectsintersection with every line intois a single line segment}}

[[File:Convex polygon illustration1.svg|right|thumb|Illustration of a convex set which looks somewhatshaped like a deformed circle. The (black) line segment joining points ''x'' and ''y'' lies completely within the (green) set, illustrated in green. Since this is true for any pointspotential xlocations andof ytwo points within the set that we might choose, the set is convex.]]

[[File:Convex polygon illustration2.svg|right|thumb|Illustration of a non-convex set. SinceThe the (red) part of the (black and red)line line-segment joining the points ''x'' and y lies ''outsidey'' partially extends outside of the (green) set, illustrated in red, and the intersection of the set iswith non-convexthe line occurs in two places, illustrated in black.]]

In [[geometry]], a subsetset of a [[Euclidean space]], or more generally an [[affine space]] over the [[Real number|reals]],points is '''convex''' if, given any two points, it contains the wholeevery [[line segment]] thatbetween joinstwo them.points Equivalently,in a '''convexthe set''' or a '''convex region''' is a subset that intersect 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|accessdateaccess-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|accessdateaccess-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]].

The notion of a convex set can be generalized as described below.

== 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(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}} and {{mvar|y}} in {{mvar|C}}, and {{mvar|t}} in the [[interval (mathematics)|interval]] {{math|[0, 1]}}. This implies that convexity (the property of being convex) is invariant under [[affine transformation]]s. This implies also that a convex set in a [[real number|real]] or [[complex number|complex]] [[topological vector space]] is [[path-connected]], thus [[connected space|connected]].

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)|interior]] of {{mvar|C}}.

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]].

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>

== Properties ==

Given {{mvar|r}} points {{math|''u''₁, ..., ''u_r''}} in a convex set {{mvar|S}}, and {{mvar|r}}

[[negative number|nonnegative number]]s {{math|''λ''₁, ..., ''λ_r''}} such that {{math|''λ''₁ + ... + ''λ_r'' {{=}} 1}}, the [[affine combination]]

:<math display=block>\sum_{k=1}^r\lambda_k u_k</math>

belongs to {{mvar|S}}. As the definition of a convex set is the case {{math|1=''r'' = 2}}, this property characterizes convex sets.

Such an affine combination is called a [[convex combination]] of {{math|''u''₁, ..., ''u_r''}}. The '''convex hull''' of a subset {{mvar|S}} of a real vector space is defined as the intersection of all convex sets that contain {{mvar|S}}. More concretely, the convex hull is the set of all convex combinations of points in {{mvar|S}}. In particular, this is a convex set.

A ''(bounded) [[convex polytope]]'' is the convex hull of a finite subset of some Euclidean space {{math|'''R'''^''n''}}.

=== Intersections and unions ===

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#The [[empty set]] and the whole space are convex.

#The intersection of any collection of convex sets is convex.

#The ''[[union (sets)|union]]'' of a sequencecollection of convex sets is convex, if theythose sets form a [[Total order#Chains|non-decreasing chain]] for(a totally ordered set) under inclusion. For this property, the restriction to chains is important, as the union of two convex sets ''need not'' be convex.

=== Closed convex sets ===

[[closed set|Closed]] convex sets are convex sets that contain all their [[limit points]]. They can be characterised as the intersections of ''closed [[Half-space (geometry)|half-space]]s'' (sets of pointpoints in space that lie on and to one side of a [[hyperplane]]).

From what has just been said, it is clear that such intersections are convex, and they will also be closed sets. To prove the converse, i.e., every closed convex set may be represented as such intersection, one needs the [[supporting hyperplane theorem]] in the form that for a given closed convex set {{mvar|C}} and point {{mvar|P}} outside it, there is a closed half-space {{mvar|H}} that contains {{mvar|C}} and not {{mvar|P}}. The supporting hyperplane theorem is a special case of the [[Hahn–Banach theorem]] of [[functional analysis]].

=== ConvexFace setsof anda rectanglesconvex set ===

A '''face''' of a convex set <math>C</math> is a convex subset <math>F</math> of <math>C</math> such that whenever a point <math>p</math> in <math>F</math> lies strictly between two points <math>x</math> and <math>y</math> in <math>C</math>, both <math>x</math> and <math>y</math> must be in <math>F</math>.{{sfn | Rockafellar| 1997 | p=162}} Equivalently, for any <math>x,y\in C</math> and any real number <math>0<t<1</math> such that <math>(1-t)x+ty</math> is in <math>F</math>, <math>x</math> and <math>y</math> must be in <math>F</math>. According to this definition, <math>C</math> itself and the empty set are faces of <math>C</math>; these are sometimes called the ''trivial faces'' of <math>C</math>. An '''[[extreme point]]''' of <math>C</math> is a point that is a face of <math>C</math>.

Let ''C'' be a [[convex body]] in the plane (a convex set whose interior is non-empty). We can inscribe a rectangle ''r'' in ''C'' such that a [[Homothetic transformation|homothetic]] copy ''R'' of ''r'' is circumscribed about ''C''. The positive homothety ratio is at most 2 and:<ref>{{Cite journal | doi = 10.1007/BF01263495| title = Approximation of convex bodies by rectangles| journal = Geometriae Dedicata| volume = 47| pages = 111–117| year = 1993| last1 = Lassak | first1 = M. | s2cid = 119508642}}</ref>

:<math>\tfrac{1}{2} \cdot\operatorname{Area}(R) \leq \operatorname{Area}(C) \leq 2\cdot \operatorname{Area}(r)</math>

<br />

Let <math>C</math> be a convex set in <math>\R^n</math> that is [[compact space|compact]] (or equivalently, closed and [[bounded set|bounded]]). Then <math>C</math> is the convex hull of its extreme points.{{sfn | Rockafellar| 1997 | p=166}} More generally, each compact convex set in a [[locally convex topological vector space]] is the closed convex hull of its extreme points (the [[Krein–Milman theorem]]).

=== Blaschke-Santaló diagrams ===

The set <math>\mathcal{K}^2</math> of all planar convex bodies can be parameterized in terms of the convex body [[Diameter#Generalizations|diameter]] ''D'', its inradius ''r'' (the biggest circle contained in the convex body) and its circumradius ''R'' (the smallest circle containing the convex body). In fact, this set can be described by the set of inequalities given by<ref name=":0">{{Cite journal|last=Santaló|first=L.|date=1961|title=Sobre los sistemas completos de desigualdades entre tres elementos de una figura convexa planas|journal=Mathematicae Notae|volume=17|pages=82–104}}</ref><ref name=":1">{{Cite journal|last1=Brandenberg|first1=René|last2=González Merino|first2=Bernardo|date=2017|title=A complete 3-dimensional Blaschke-Santaló diagram|url=http://mia.ele-math.com/20-22|journal=Mathematical Inequalities & Applications|language=en|issue=2|pages=301–348|doi=10.7153/mia-20-22|issn=1331-4343|doi-access=free}}</ref><blockquote><math>2r \le D \le 2R</math>

For example:

<math>R \le \frac{\sqrt{3}}{3} D</math>

* A [[triangle]] in the plane (including the region inside) is a compact convex set. Its nontrivial faces are the three vertices and the three edges. (So the only extreme points are the three vertices.)

* The only nontrivial faces of the [[closed unit disk]] <math>\{ (x,y) \in \R^2: x^2+y^2 \leq 1 \}</math> are its extreme points, namely the points on the [[unit circle]] <math>S^1 = \{ (x,y) \in \R^2: x^2+y^2=1 \}</math>.

=== Convex sets and rectangles ===

<math>r + R \le D</math>

Let ''{{mvar|C''}} be a [[convex body]] in the plane (a convex set whose interior is non-empty). We can inscribe a rectangle ''r'' in ''{{mvar|C''}} such that a [[Homothetic transformation|homothetic]] copy ''R'' of ''r'' is circumscribed about ''{{mvar|C''}}. The positive homothety ratio is at most 2 and:<ref>{{Cite journal | doi = 10.1007/BF01263495| title = Approximation of convex bodies by rectangles| journal = Geometriae Dedicata| volume = 47| pages = 111–117| year = 1993| last1 = Lassak | first1 = M. | s2cid = 119508642}}</ref>

:<math display=block>\tfrac{1}{2} \cdot\operatorname{Area}(R) \leq \operatorname{Area}(C) \leq 2\cdot \operatorname{Area}(r)</math>

<br />

=== Blaschke-Santaló diagrams ===

<math>D^2 \sqrt{4R^2-D^2} \le 2R (2R + \sqrt{4R^2 -D^2})</math></blockquote>and can be visualized as the image of the function ''g'' that maps a convex body to the {{math|'''R'''²}} point given by (''r''/''R'', ''D''/2''R''). The image of this function is known a (''r'', ''D'', ''R'') Blachke-Santaló diagram.<ref name=":1" />

The set <math>\mathcal{K}^2</math> of all planar convex bodies can be parameterized in terms of the convex body [[Diameter#Generalizations of a set|diameter]] ''D'', its inradius ''r'' (the biggest circle contained in the convex body) and its circumradius ''R'' (the smallest circle containing the convex body). In fact, this set can be described by the set of inequalities given by<ref name=":0">{{Cite journal|last=Santaló|first=L.|date=1961|title=Sobre los sistemas completos de desigualdades entre tres elementos de una figura convexa planas|journal=Mathematicae Notae|volume=17|pages=82–104}}</ref><ref name=":1">{{Cite journal|last1=Brandenberg|first1=René|last2=González Merino|first2=Bernardo|date=2017|title=A complete 3-dimensional Blaschke-Santaló diagram|url=http://mia.ele-math.com/20-22|journal=Mathematical Inequalities & Applications|language=en|issue=2|pages=301–348|doi=10.7153/mia-20-22|issn=1331-4343|doi-access=free|arxiv=1404.6808}}</ref><blockquote><math>2r \le D \le 2R</math>

<math display=block>2r \le D \le 2R</math>

<math display=block>R \le \frac{\sqrt{3}}{3} D</math>

<math display=block>r + R \le D</math>

<math display=block>D^2 \sqrt{4R^2-D^2} \le 2R (2R + \sqrt{4R^2 -D^2})</math>

<math>D^2 \sqrt{4R^2-D^2} \le 2R (2R + \sqrt{4R^2 -D^2})</math></blockquote>and can be visualized as the image of the function ''g'' that maps a convex body to the {{math|'''R'''²}} point given by (''r''/''R'', ''D''/2''R''). The image of this function is known a (''r'', ''D'', ''R'') Blachke-Santaló diagram.<ref name=":1" />

[[File:Blaschke-Santaló_diagram_for_planar_convex_bodies.pdf|alt=|center|thumb|673x673px|Blaschke-Santaló (''r'', ''D'', ''R'') diagram for planar convex bodies. <math>\mathbb{L}</math> denotes the line segment, <math>\mathbb{I}_{\frac{\pi}{3}}</math> the equilateral triangle, <math>\mathbb{RT}</math> the [[Reuleaux triangle]] and <math>\mathbb{B}_2</math> the unit circle.]]

Alternatively, the set <math>\mathcal{K}^2</math> can also be parametrized by its width (the smallest distance between any two different parallel support hyperplanes), perimeter and area.<ref name=":0" /><ref name=":1" />

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=== Convex hulls ===

{{Main|convex hull}}

Every subset {{mvar|A}} of the vector space is contained within a smallest convex set (called the [[''convex hull]]'' of {{mvar|A}}), namely the intersection of all convex sets containing {{mvar|A}}. The convex-hull operator Conv() has the characteristic properties of a [[closure operator|hull operator]]:

|* ''extensive'': {{math|''S''&nbsp;⊆&nbsp;Conv(''S'')}},

:{| border="0"

* ''[[Monotone function#Monotonicity in order theory|non-decreasing]]'': {{math|''S''&nbsp;⊆&nbsp;''T''}} implies that {{math|Conv(''S'')&nbsp;⊆&nbsp;Conv(''T'')}}, and

|-

* ''[[idempotence|idempotent]]'': {{math|Conv(Conv(''S'')) {{=}} Conv(''S'')}}.

| ''extensive''

| {{math|''S''&nbsp;⊆&nbsp;Conv(''S'')}},

|-

| ''[[Monotone function#Monotonicity in order theory|non-decreasing]]''

| {{math|''S''&nbsp;⊆&nbsp;''T''}} implies that {{math|Conv(''S'')&nbsp;⊆&nbsp;Conv(''T'')}}, and

|-

| ''[[idempotence|idempotent]]''

| {{math|Conv(Conv(''S'')) {{=}} Conv(''S'')}}.

|}

The convex-hull operation is needed for the set of convex sets to form a <!-- complete -->[[lattice (order)|lattice]], in which the [[join and meet|"''join''" operation]] is the convex hull of the union of two convex sets

:<math display=block>\operatorname{{math|Conv}(''S'') ∨ \vee\operatorname{Conv}(''T'') {{=}} \operatorname{Conv}(''S''&nbsp;∪&nbsp;''\cup T'') {{=}} \operatorname{Conv}\bigl(\operatorname{Conv}(''S'')&nbsp;∪&nbsp;\cup\operatorname{Conv}(''T'')\bigr)}}.</math>

The intersection of any collection of convex sets is itself convex, so the convex subsets of a (real or complex) vector space form a complete [[lattice (order)|lattice]].

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In a real vector-space, the ''[[Minkowski addition|Minkowski sum]]'' of two (non-empty) sets, {{math|''S''₁}} and {{math|''S''₂}}, is defined to be the [[sumset|set]] {{math|''S''₁&nbsp;+&nbsp;''S''₂}} formed by the addition of vectors element-wise from the summand-sets

<math display=block>S_1+S_2=\{x_1+x_2: x_1\in S_1, x_2\in S_2\}.</math>

:{{math|''S''₁&nbsp;+&nbsp;''S''₂ {{=}} {''x''₁&nbsp;+&nbsp;''x''₂ : ''x''₁&nbsp;∈&nbsp;''S''₁, ''x''₂&nbsp;∈&nbsp;''S''₂} }}.

More generally, the ''Minkowski sum'' of a finite family of (non-empty) sets {{math|''S_n''}} is <!-- defined to be --> the set <!-- of vectors --> formed by element-wise addition of vectors<!-- from the summand-sets -->

:<math display=block> \sum_n S_n = \left \{ \sum_n x_n : x_n \in S_n \right \}.</math>

For Minkowski&nbsp;addition, the ''zero set''&nbsp;{{math|{0} }} containing only the [[null vector|zero&nbsp;vector]]&nbsp;{{math|0}} has [[identity element|special importance]]: For every non-empty subset&nbsp;S of a vector space

<math display=block>S+\{0\}=S;</math>

:{{math|''S''&nbsp;+&nbsp;{0} {{=}} ''S''}};

in algebraic terminology, {{math|{0} }} is the [[identity element]] of Minkowski addition (on the collection of non-empty sets).<ref>The [[empty set]] is important in Minkowski addition, because the empty set annihilates every other subset: For every subset {{mvar|S}} of a vector space, its sum with the empty set is empty: {{<math|''>S'' + ∅ {{\emptyset=}} ∅}}\emptyset</math>.</ref>

=== Convex hulls of Minkowski sums ===

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Let {{math|''S''₁, ''S''₂}} be subsets of a real vector-space, the [[convex hull]] of their Minkowski sum is the Minkowski sum of their convex hulls

<math display=block>\operatorname{Conv}(S_1+S_2)=\operatorname{Conv}(S_1)+\operatorname{Conv}(S_2).</math>

:{{math|Conv(''S''₁&nbsp;+&nbsp;''S''₂) {{=}} Conv(''S''₁) + Conv(''S''₂)}}.

This result holds more generally for each finite collection of non-empty sets:

:<math display=block>\text{Conv}\left ( \sum_n S_n \right ) = \sum_n \text{Conv} \left (S_n \right).</math>

In mathematical terminology, the [[operation (mathematics)|operation]]s of Minkowski summation and of forming [[convex hull]]s are [[commutativity|commuting]] operations.<ref>Theorem&nbsp;3 (pages&nbsp;562–563): {{cite newsjournal|first1=M.|last1=Krein|author-link1=Mark Krein|first2=V.|last2=Šmulian|year=1940|title=On regularly convex sets in the space conjugate to a Banach space|journal=Annals of Mathematics |series=Second Series| volume=41 |issue=3 |pages=556–583|jstor=1968735|doi=10.2307/1968735}}</ref><ref name="Schneider">For the commutativity of [[Minkowski addition]] and [[convex hull|convexification]], see Theorem&nbsp;1.1.2 (pages&nbsp;2–3) in Schneider; this reference discusses much of the literature on the [[convex hull]]s of [[Minkowski addition|Minkowski]] [[sumset]]s in its "Chapter&nbsp;3 Minkowski addition" (pages&nbsp;126–196): {{cite book|last=Schneider|first=Rolf|title=Convex bodies: The Brunn–Minkowski theory|series=Encyclopedia of mathematics and its applications|volume=44|publisher=Cambridge&nbsp;University Press|___location=Cambridge|year=1993|pages=xiv+490|isbn=0-521-35220-7|mr=1216521|url=https://archive.org/details/convexbodiesbrun0000schn}}</ref>

:<math>\text{Conv}\left ( \sum_n S_n \right ) = \sum_n \text{Conv} \left (S_n \right).</math>

In mathematical terminology, the [[operation (mathematics)|operation]]s of Minkowski summation and of forming [[convex hull]]s are [[commutativity|commuting]] operations.<ref>Theorem&nbsp;3 (pages&nbsp;562–563): {{cite news|first1=M.|last1=Krein|author-link1=Mark Krein|first2=V.|last2=Šmulian|year=1940|title=On regularly convex sets in the space conjugate to a Banach space|journal=Annals of Mathematics |series=Second Series| volume=41 |pages=556–583|jstor=1968735|doi=10.2307/1968735}}</ref><ref name="Schneider">For the commutativity of [[Minkowski addition]] and [[convex hull|convexification]], see Theorem&nbsp;1.1.2 (pages&nbsp;2–3) in Schneider; this reference discusses much of the literature on the [[convex hull]]s of [[Minkowski addition|Minkowski]] [[sumset]]s in its "Chapter&nbsp;3 Minkowski addition" (pages&nbsp;126–196): {{cite book|last=Schneider|first=Rolf|title=Convex bodies: The Brunn–Minkowski theory|series=Encyclopedia of mathematics and its applications|volume=44|publisher=Cambridge&nbsp;University Press|___location=Cambridge|year=1993|pages=xiv+490|isbn=0-521-35220-7|mr=1216521|url=https://archive.org/details/convexbodiesbrun0000schn}}</ref>

=== Minkowski sums of convex sets ===

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The following famous theorem, proved by Dieudonné in 1966, gives a sufficient condition for the difference of two closed convex subsets to be closed.<ref name="Zalinescu p. 7">{{cite book|last=Zălinescu|first=C.|title=Convex analysis in general vector spaces|url=https://archive.org/details/convexanalysisge00zali_934|url-access=limited|publisher=World Scientific Publishing Co., Inc|___location= River Edge, NJ |date= 2002|page=[https://archive.org/details/convexanalysisge00zali_934/page/n27 7]|isbn=981-238-067-1|mr=1921556}}</ref> It uses the concept of a '''recession cone''' of a non-empty convex subset ''S'', defined as:

: <math display=block>\operatorname{rec} S = \left\{ x \in X \, : \, x + S \subseteq S \right\},</math>.

where this set is a [[convex cone]] containing <math>0 \in X </math> and satisfying <math>S + \operatorname{rec} S = S</math>. Note that if ''S'' is closed and convex then <math>\operatorname{rec} S</math> is closed and for all <math>s_0 \in S</math>,
<math display=block>\operatorname{rec} S = \bigcap_{t > 0} t (S - s_0).</math>.

'''Theorem''' (Dieudonné). Let ''A'' and ''B'' be non-empty, closed, and convex subsets of a [[locally convex topological vector space]] such that <math>\operatorname{rec} A \cap \operatorname{rec} B</math> is a linear subspace. If ''A'' or ''B'' is [[locally compact]] then ''A''&nbsp;−&nbsp;''B'' is closed.

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Let {{math|''Y'' ⊆ ''X''}}. The subspace {{mvar|Y}} is a convex set if for each pair of points {{math|''a'', ''b''}} in {{mvar|Y}} such that {{math|''a'' ≤ ''b''}}, the interval {{math|[''a'', ''b''] {{=}} {''x'' ∈ ''X'' {{!}} ''a'' ≤ ''x'' ≤ ''b''} }} is contained in {{mvar|Y}}. That is, {{mvar|Y}} is convex if and only if for all {{math|''a'', ''b''}} in {{mvar|Y}}, {{math|''a'' ≤ ''b''}} implies {{math|[''a'', ''b''] ⊆ ''Y''}}.

A convex set is '''{{em|not'''}} connected in general: a counter-example is given by the spacesubspace {1,2,3} in {{math|'''QZ'''}}, which is both convex and [[totallynot disconnected]]connected.

=== Convexity spaces ===

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Given a set {{mvar|X}}, a '''convexity''' over {{mvar|X}} is a collection {{math|''𝒞''}} of subsets of {{mvar|X}} satisfying the following axioms:<ref name="Soltan"/><ref name="Singer"/><ref name="vanDeVel" >{{cite book|last=van De Vel|first=Marcel L. J.|title=Theory of convex structures|series=North-Holland Mathematical Library|publisher=North-Holland Publishing Co.|___location=Amsterdam|year= 1993|pages=xvi+540|isbn=0-444-81505-8|mr=1234493}}</ref>

#The empty set and {{mvar|X}} are in {{math|''𝒞''}}.

#The intersection of any collection from {{math|''𝒞''}} is in {{math|''𝒞''}}.

#The union of a [[Total order|chain]] (with respect to the [[inclusion relation]]) of elements of {{math|''𝒞''}} is in {{math|''𝒞''}}.

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For an alternative definition of abstract convexity, more suited to [[discrete geometry]], see the ''convex geometries'' associated with [[antimatroid]]s.

=== Convex spaces ===

{{main|Convex space}}

Convexity can be generalised as an abstract algebraic structure: a space is convex if it is possible to take convex combinations of points.

== See also ==

{{Div col|colwidth=25em}}

* [[Absorbing set]]

* [[Algorithmic problems on convex sets]]

* [[Bounded set (topological vector space)]]

* [[Brouwer fixed-point theorem]]

* [[Complex convexity]]

* [[Convex hullcone]]

* [[Convex series]]

* [[Convex metric space]]

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* [[Holomorphically convex hull]]

* [[Integrally-convex set]]

* [[John ellipsoid]]

* [[Pseudoconvexity]]

* [[Radon's theorem]]

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{{reflist|30em}}

==Bibliography==

== External links ==

* {{cite book | last=Rockafellar | first=R. T. | author-link=R. Tyrrell Rockafellar | title=Convex Analysis |publisher=Princeton University Press | ___location=Princeton, NJ | orig-year=1970 | year=1997 | isbn=1-4008-7317-7 |url=https://books.google.com/books?id=1TiOka9bx3sC }}

== External links ==

{{Wiktionary}}

* {{springer|title=Convex subset|id=p/c026380|mode=cs1}}

* [http://www.fmf.uni-lj.si/~lavric/lauritzen.pdf Lectures on Convex Sets], notes by Niels Lauritzen, at [[Aarhus University]], March 2010.

{{Functional Analysisanalysis}}

{{Convex analysis and variational analysis}}

{{Authority control}}

{{DEFAULTSORT:Convex Set}}

[[Category:Convex geometry]]

[[Category:Convex analysis]]

[[Category:Convex geometry]]