Closed convex sets are convex sets that contain all their limit points. They can be characterised as the intersections of closed half-spaces (sets of point in space that lie on and to one side of a hyperplane).

Are convex hulls closed?

The closed convex hull of a set is the closure of the convex hull, and the open convex hull is the interior (or in some sources the relative interior) of the convex hull. ... However, an intersection of closed half-spaces is itself closed, so when a convex hull is not closed it cannot be represented in this way.

How do you prove a convex set is closed?

Indeed, any closed convex set is the intersection of all halfspaces that contain it: C = ∩{H|Hhalfspaces,C ⊆ H}. A standard way to prove that a set (or later, a function) is convex is to build it up from simple sets for which convexity is known, by using convexity preserving operations.