Editing Convex set (section)

== Convex hulls and Minkowski sums ==

=== 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]]:
* ''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{Conv}(S)\vee\operatorname{Conv}(T) = \operatorname{Conv}(S\cup T) = \operatorname{Conv}\bigl(\operatorname{Conv}(S)\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]].

=== Minkowski addition ===
{{Main|Minkowski addition}}
[[File:Minkowski sum graph - vector version.svg|thumb|alt=Three squares are shown in the nonnegative quadrant of the Cartesian plane. The square {{math|''Q''<sub>1</sub> {{=}} [0, 1] × [0, 1]}} is green. The square {{math|''Q''<sub>2</sub> {{=}} [1, 2] × [1, 2]}} is brown, and it sits inside the turquoise square {{math|1=Q<sub>1</sub>+Q<sub>2</sub>{{=}}[1,3]×[1,3]}}.|[[Minkowski addition]] of sets. The <!-- [[Minkowski addition|Minkowski]]&nbsp; -->[[sumset|sum]] of the squares&nbsp;Q<sub>1</sub>=[0,1]<sup>2</sup> and&nbsp;Q<sub>2</sub>=[1,2]<sup>2</sup> is the square&nbsp;Q<sub>1</sub>+Q<sub>2</sub>=[1,3]<sup>2</sup>.]]

In a real vector-space, the ''[[Minkowski addition|Minkowski sum]]'' of two (non-empty) sets, {{math|''S''<sub>1</sub>}} and {{math|''S''<sub>2</sub>}}, is defined to be the [[sumset|set]] {{math|''S''<sub>1</sub>&nbsp;+&nbsp;''S''<sub>2</sub>}} 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>
More generally, the ''Minkowski sum'' of a finite family of (non-empty) sets {{math|''S<sub>n</sub>''}} 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>
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 ===
Minkowski addition behaves well with respect to the operation of taking convex hulls, as shown by the following proposition:

Let {{math|''S''<sub>1</sub>, ''S''<sub>2</sub>}} 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>

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 journal|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>

=== Minkowski sums of convex sets ===
The Minkowski sum of two compact convex sets is compact. The sum of a compact convex set and a closed convex set is closed.<ref>Lemma&nbsp;5.3: {{cite book|first1=C.D.|last1= Aliprantis|first2=K.C.| last2=Border|title=Infinite Dimensional Analysis, A Hitchhiker's Guide| publisher=Springer| location=Berlin|year=2006|isbn=978-3-540-29587-7}}</ref>

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.