{{Multiple issues| {{one source|date=December 2024}} {{Context|date=December 2024}} }} [[File:Extremenotexposed.png|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.]]
In [[mathematics]], most commonly in [[convex geometry]], an '''extreme set''' or '''face''' of a set <math>C\subseteq V</math> in a [[vector space]] <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[0,1]</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 '''[[extreme point]]''' 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 '''[[exposed face]]''' 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 '''[[exposed point]]''' 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. 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 == 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 [[Convex set|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 <math>f</math> to be continuous in a given [[vector topology]].
==See also== * [[Face (geometry)]]
==References== {{reflist}}
==Bibliography== * {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!--{{sfn|Narici|Beckenstein|2011|p=}}-->
==External links== <!-- These sources contain little information. Replace by better ones when found.--> * [https://spot.colorado.edu/~baggett/funcchap3.pdf TOPOLOGICAL VECTOR SPACES AND CONTINUOUS LINEAR FUNCTIONALS], Chapter III of FUNCTIONAL ANALYSIS, Lawrence Baggett, University of Colorado Boulder. * [https://www.math.lmu.de/~philip/publications/lectureNotes/philipPeter_FunctionalAnalysis.pdf Functional Analysis], Peter Philip, Ludwig-Maximilians-universität München, 2024 [[Category:Convex geometry]] [[Category:Convex hulls]] [[Category:Functional analysis]] [[Category:Mathematical analysis]]