Editing Convex set (section)

== Properties ==
Given {{mvar|r}} points {{math|''u''<sub>1</sub>, ..., ''u<sub>r</sub>''}} in a convex set {{mvar|S}}, and {{mvar|r}}
[[negative number|nonnegative number]]s {{math|''λ''<sub>1</sub>, ..., ''λ<sub>r</sub>''}} such that {{math|''λ''<sub>1</sub> + ... + ''λ<sub>r</sub>'' {{=}} 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''<sub>1</sub>, ..., ''u<sub>r</sub>''}}. 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'''<sup>''n''</sup>}}.

=== Intersections and unions ===
The collection of convex subsets of a vector space, an affine space, or a [[Euclidean space]]  has the following properties:<ref name="Soltan" >Soltan, Valeriu, ''Introduction to the Axiomatic Theory of Convexity'', Ştiinţa, [[Chişinău]], 1984 (in Russian).
</ref><ref name="Singer" >{{cite book|last=Singer|first=Ivan|title=Abstract convex analysis|series=Canadian Mathematical Society series of monographs and advanced texts|publisher=John Wiley&nbsp;&&nbsp;Sons, Inc.|location=New&nbsp;York|year= 1997|pages=xxii+491|isbn=0-471-16015-6|mr=1461544}}</ref>
#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 collection of convex sets is convex if those sets form a [[Total order#Chains|chain]] (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 points 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]].

=== Face of a convex 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 <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]]).

For example:
* 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 ===
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 ===
The set  <math>\mathcal{K}^2</math>  of all planar convex bodies can be parameterized in terms of the convex body [[Diameter 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>
<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>
and can be visualized as the image of the function ''g'' that maps a convex body to the {{math|'''R'''<sup>2</sup>}} 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" />

=== Other properties ===
Let ''X'' be a topological vector space and <math>C \subseteq X</math> be convex. 
* <math>\operatorname{Cl} C</math> and <math>\operatorname{Int} C</math> are both convex (i.e. the closure and interior of convex sets are convex).
* If <math>a \in \operatorname{Int} C</math> and <math>b \in \operatorname{Cl} C</math> then <math>[a, b[ \, \subseteq \operatorname{Int} C</math> (where <math>[a, b[ \, := \left\{ (1 - r) a + r b : 0 \leq r < 1 \right\}</math>).
* If <math>\operatorname{Int} C \neq \emptyset</math> then:
** <math>\operatorname{cl} \left( \operatorname{Int} C \right) = \operatorname{Cl} C</math>, and
** <math>\operatorname{Int} C = \operatorname{Int} \left( \operatorname{Cl} C \right) = C^i</math>, where <math>C^{i}</math> is the [[algebraic interior]] of ''C''.