Undid revision 1273774098 by BTotaro (talk) are you sure that's the definition of a face? It would appear to define faces to equal all convex subsets of a convex set (e.g. line segments interior to a triangle).
From what has just been said, it is clear that such intersections are convex, and they will also be closed sets. To prove the converse, i.e., every closed convex set may be represented as such intersection, one needs the [[supporting hyperplane theorem]] in the form that for a given closed convex set {{mvar|C}} and point {{mvar|P}} outside it, there is a closed half-space {{mvar|H}} that contains {{mvar|C}} and not {{mvar|P}}. The supporting hyperplane theorem is a special case of the [[Hahn–Banach theorem]] of [[functional analysis]].
=== Face of a convex set ===
A '''face''' of a convex set <math>C</math> is a subset <math>F</math> of <math>C</math> such that <math>F</math> is also a convex set, and for any points <math>x,y</math> in <math>C</math> and any real number <math>0<t<1</math> with <math>(1-t)x+ty</math> in <math>F</math>, <math>x</math> and <math>y</math> must both be in <math>F</math>.{{sfn | Rockafellar| 1997 | p=162}} For example, <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}} This holds more generally for a compact convex set in any [[locally convex topological vector space]] (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 extremal points are the three vertices.) For another example: the only nontrivial faces of the [[closed unit disk]] <math>\{ (x,y) \in \R^2: x^2+y^2 \leq 1 \}</math> are its extremal points, namely the points on the [[unit circle]] <math>S^1 = \{ (x,y) \in \R^2: x^2+y^2=1 \}</math>.
