Convex set: Difference between revisions

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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 in Euclidean spaces can be generalized in several ways by modifying its definition, for instance by restricting the line segments that such a set is required to contain.

== Definitions ==

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== 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]]

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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 ===

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=== 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> islies alsostrictly abetween convextwo set,points <math>x</math> and for<math>y</math> anyin points<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> withsuch that <math>(1-t)x+ty</math> is in <math>F</math>, <math>x</math> and <math>y</math> must both be in <math>F</math>.{{sfn |According Rockafellar| 1997 | p=162}}to Forthis exampledefinition, <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]]).

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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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{{reflist|30em}}

==SourcesBibliography==

* {{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 }}