Content deleted Content added
m Minor correction |
rm equivalent per talk |
||
| (One intermediate revision by one other user 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 set of points is '''convex''' if it contains every [[line segment]] between two points in the set
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 31:
== 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 46 ⟶ 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 54 ⟶ 53:
=== 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>
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]]).
Line 159 ⟶ 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 197 ⟶ 196:
{{reflist|30em}}
==
* {{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 }}
| |||