Editing Convex set (section)

=== Closed convex sets ===
[[closed set|Closed]] convex sets are convex sets that contain all their [[limit points]]. They can be characterised as the intersections of ''closed [[Half-space (geometry)|half-space]]s'' (sets of points in space that lie on and to one side of a [[hyperplane]]).

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]].