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Revision as of 10:21, 20 December 2024 editRigmat (talk | contribs)52 edits ←Created page with 'thumb|The two distinguished points are examples of extreme points of a convex set that are not exposed points. Therefore, not every convex face of a convex set is an exposed face. Let <math>C\subseteq V</math>, where <math>V</math> is a vector space. A '''extreme set''' or '''face''' or of <math>C</math> is a set <math>F\subseteq C</math> such that <math>x,y\in C \ \&\ 0<\theta<1 \ \&\ \theta x+(1-\theta)y\in F\ \Right...'Tag: Disambiguation links added  Latest revision as of 17:43, 2 January 2025 edit undoMWinter4 (talk | contribs)Extended confirmed users602 edits Improved introduction; removed short section on facts and moved content to introduction 
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Let <math>C\subseteq V</math>, where <math>V</math> is a ].


A '''extreme set''' or '''face''' or of <math>C</math> is a set <math>F\subseteq C</math> such that <math>x,y\in C \ \&\ 0<\theta<1 \ \&\ \theta x+(1-\theta)y\in F\ \Rightarrow\ x,y\in F</math>.{{sfn|Narici|Beckenstein|2011|pp=275-339}} That is, if a point <math>p\in F</math> lies between some points <math>x,y\in C</math>, then <math>x,y\in F</math>. In ], most commonly in ], an '''extreme set''' or '''face''' of a set <math>C\subseteq V</math> in a ] <math>V</math> is a subset <math>F\subseteq C</math> with the property that if for any two points <math>x,y\in C</math> some in-between point <math>z=\theta x + (1-\theta) y,\theta\in</math> lies in <math>F</math>, then we must have had <math>x,y\in F</math>.{{sfn|Narici|Beckenstein|2011|pp=275-339}}
<!--That is, if a point <math>p\in F</math> lies between some points <math>x,y\in C</math>, then <math>x,y\in F</math>.-->
An ''']''' of <math>C</math> is a point <math>p\in C</math> for which <math>\{p\}</math> is a face.{{sfn|Narici|Beckenstein|2011|pp=275-339}}
<!--That is, if <math>p</math> lies between some points <math>x,y\in C</math>, then <math>x=y=p</math>.-->


An ''']''' of <math>C</math> is a point <math>p\in C</math> such that <math>\{p\}</math> is a face of <math>C</math>.{{sfn|Narici|Beckenstein|2011|pp=275-339}} That is, if <math>p</math> lies between some points <math>x,y\in C</math>, then <math>x=y=p</math>. An ''']''' of <math>C</math> is the subset of points of <math>C</math> where a linear functional achieves its minimum on <math>C</math>. Thus, if <math>f</math> is a linear functional on <math>V</math> and <math>\alpha =\inf\{ f(c)\ \colon c\in C\}>-\infty</math>, then <math> \{c\in C\ \colon f(c)=\alpha\}</math> is an exposed face of <math>C</math>.
⚫ An ''']''' of <math>C</math> is a point <math>p\in C</math> such that <math>\{p\}</math> is an exposed face. That is, <math>f(p) > f(c)</math> for all <math>c\in C\setminus\{p\}</math>.


⚫ An exposed face is a face, but the converse is not true (see the figure). An exposed face of <math>C</math> is convex if <math>C</math> is convex.
An ''']''' of <math>C</math> is the subset of points of <math>C</math> where a linear functional achieves its minimum on <math>C</math>. Thus, if <math>f</math> is a linear functional on <math>V</math> and <math>\alpha =\inf\{ fc\ \colon c\in C\}>-\infty</math>, then <math> \{c\in C\ \colon fc=\alpha\}</math> is an exposed face of <math>C</math>.
⚫ If <math>F</math> is a face of <math>C\subseteq V</math>, then <math>E\subseteq F</math> is a face of <math>F</math> if and only if <math> E</math> is a face of <math> C</math>.


⚫ == Competing definitions ==
⚫ An ''']''' of <math>C</math> is a point <math>p\in C</math> such that <math>\{p\}</math> is an exposed face of <math>C</math>. That is, <math>fp > fc</math> for all <math>c\in C\setminus\{p\}</math>.

⚫ === Competing definitions ===
Some authors do not include <math>C</math> and/or <math>\varnothing</math> among the (exposed) faces. Some authors require <math>F</math> and/or <math>C</math> to be ] (else the boundary of a disc is a face of the disc, as well as any subset of the boundary) or closed. Some authors require the functional <math>f</math> to be continuous in a given ]. Some authors do not include <math>C</math> and/or <math>\varnothing</math> among the (exposed) faces. Some authors require <math>F</math> and/or <math>C</math> to be ] (else the boundary of a disc is a face of the disc, as well as any subset of the boundary) or closed. Some authors require the functional <math>f</math> to be continuous in a given ].

=== Facts ===
⚫ An exposed face is clearly a face. An exposed face of <math>C</math> is clearly convex if <math>C</math> is convex.

⚫ If <math>F</math> is a face of <math>C\subseteq V</math>, then <math>E\subseteq F</math> is a face of <math>F</math> iff <math> E</math> is a face of <math> C</math>.


==See also== ==See also==

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The two distinguished points are examples of extreme points of a convex set that are not exposed points. Therefore, not every convex face of a convex set is an exposed face.

In mathematics, most commonly in convex geometry, an extreme set or face of a set C ⊆ V {\displaystyle C\subseteq V} in a vector space V {\displaystyle V} is a subset F ⊆ C {\displaystyle F\subseteq C} with the property that if for any two points x , y ∈ C {\displaystyle x,y\in C} some in-between point z = θ x + ( 1 − θ ) y , θ ∈ [ 0 , 1 ] {\displaystyle z=\theta x+(1-\theta )y,\theta \in } lies in F {\displaystyle F} , then we must have had x , y ∈ F {\displaystyle x,y\in F} . An extreme point of C {\displaystyle C} is a point p ∈ C {\displaystyle p\in C} for which { p } {\displaystyle \{p\}} is a face.

An exposed face of C {\displaystyle C} is the subset of points of C {\displaystyle C} where a linear functional achieves its minimum on C {\displaystyle C} . Thus, if f {\displaystyle f} is a linear functional on V {\displaystyle V} and α = inf { f ( c )   : c ∈ C } > − ∞ {\displaystyle \alpha =\inf\{f(c)\ \colon c\in C\}>-\infty } , then { c ∈ C   : f ( c ) = α } {\displaystyle \{c\in C\ \colon f(c)=\alpha \}} is an exposed face of C {\displaystyle C} . An exposed point of C {\displaystyle C} is a point p ∈ C {\displaystyle p\in C} such that { p } {\displaystyle \{p\}} is an exposed face. That is, f ( p ) > f ( c ) {\displaystyle f(p)>f(c)} for all c ∈ C ∖ { p } {\displaystyle c\in C\setminus \{p\}} .

An exposed face is a face, but the converse is not true (see the figure). An exposed face of C {\displaystyle C} is convex if C {\displaystyle C} is convex. If F {\displaystyle F} is a face of C ⊆ V {\displaystyle C\subseteq V} , then E ⊆ F {\displaystyle E\subseteq F} is a face of F {\displaystyle F} if and only if E {\displaystyle E} is a face of C {\displaystyle C} .

Competing definitions

Some authors do not include C {\displaystyle C} and/or ∅ {\displaystyle \varnothing } among the (exposed) faces. Some authors require F {\displaystyle F} and/or C {\displaystyle C} to be convex (else the boundary of a disc is a face of the disc, as well as any subset of the boundary) or closed. Some authors require the functional f {\displaystyle f} to be continuous in a given vector topology.

See also

References

  1. ^ Narici & Beckenstein 2011, pp. 275–339.

Bibliography

  • Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.

External links

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