Content deleted Content added
m Typo fix |
rm equivalent per talk |
||
| (12 intermediate revisions by 5 users not shown) | |||
Line 3:
[[File:Convex polygon illustration2.svg|right|thumb|Illustration of a non-convex set. The line segment joining points ''x'' and ''y'' partially extends outside of the set, illustrated in red, and the intersection of the set with the line occurs in two places, illustrated in black.]]
In [[geometry]], a
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.
Line 10:
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]].
== Definitions ==
Line 26:
=== 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''<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]]
Line 37 ⟶ 36:
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 ===
Line 44 ⟶ 45:
#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
=== Closed convex sets ===
Line 50 ⟶ 51:
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 ===
Line 57 ⟶ 67:
=== 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
<math display=block>2r \le D \le 2R</math>
<math display=block>R \le \frac{\sqrt{3}}{3} D</math>
Line 78 ⟶ 88:
=== Convex hulls ===
{{Main|convex hull}}
Every subset {{mvar|A}} of the vector space is contained within a smallest convex set (called the
* ''extensive'': {{math|''S'' ⊆ Conv(''S'')}},
* ''[[Monotone function#Monotonicity in order theory|non-decreasing]]'': {{math|''S'' ⊆ ''T''}} implies that {{math|Conv(''S'') ⊆ Conv(''T'')}}, and
Line 148 ⟶ 158:
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''}}.
Line 168 ⟶ 178:
* [[Brouwer fixed-point theorem]]
* [[Complex convexity]]
* [[Convex
* [[Convex series]]
* [[Convex metric space]]
Line 185 ⟶ 195:
== References ==
{{reflist|30em}}
==Bibliography==
* {{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==
| |||