Editing Face (geometry) (section)

==Face of a convex set==
[[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 face of a convex set is an exposed face.]]
The notion of a face can be generalized from convex polytopes to all [[convex set]]s, as follows. Let <math>C</math> be a convex set in a real [[vector space]] <math>V</math>. A '''face''' of <math>C</math> is a convex subset <math>F\subseteq C</math> such that whenever a point <math>p\in 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>. Equivalently, for any <math>x,y\in C</math> and any real number <math>0<\theta<1</math> such that <math>\theta x+(1-\theta)y</math> is in <math>F</math>, <math>x</math> and <math>y</math> must be in <math>F</math>.{{sfn|Rockafellar|1997|p=162}}

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 <math>p\in C</math> such that <math>\{p\}</math> is a face of <math>C</math>.{{sfn|Rockafellar|1997|p=162}} That is, if <math>p</math> lies between two points <math>x,y\in C</math>, then <math>x=y=p</math>.

For example:
* A [[triangle]] in the plane (including the region inside) is a 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>.

Let <math>C</math> be a convex set in <math>\R^n</math> that is [[compact space|compact]] (or equivalently, [[closed set|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]]).

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 of <math>C</math>. That is, <math>f(p) > f(c)</math> for all <math>c\in C\setminus\{p\}</math>. See the figure for examples of extreme points that are not exposed.

=== Competing definitions ===
Some authors do not include <math>C</math> and/or <math>\varnothing</math> as faces of <math>C</math>. Some authors require a face to be a closed subset; this is automatic for <math>C</math> a compact convex set in a vector space of finite dimension, but not in infinite dimensions.<ref>{{cite book | last=Simon | first=Barry | author-link=Barry Simon | title=Convexity: an Analytic Viewpoint |mr=2814377|publisher=Cambridge University Press | location=Cambridge | year=2011 | page=123|isbn=978-1-107-00731-4 |url=https://books.google.com/books?id=xWCs0lWGxjkC }}</ref> In infinite dimensions, the functional <math>f</math> is usually assumed to be continuous in a given [[vector topology]].

=== Properties ===
An exposed face of a convex set is a face. In particular, it is a convex subset.

If <math>F</math> is a face of a convex set <math>C</math>, then a subset <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>.