Editing Convex set (section)

=== 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> 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>.{{sfn | Rockafellar| 1997 | p=162}} Equivalently, for any <math>x,y\in C</math> and any real number <math>0<t<1</math> such that <math>(1-t)x+ty</math> is in <math>F</math>, <math>x</math> and <math>y</math> must be in <math>F</math>. 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 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]]).

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