the convex hull of its extreme points. However, this may not be true for convex sets that are not compact; for instance, the whole Euclidean plane and the open unit ball are both convex, but neither one has any extreme points. Choquet theory extends this theory from finite convex combinations of extreme points to infinite combinations (integrals) in more general spaces. Geometric and algebraic properties Closure operator The convex-hull operator has the characteristic properties of a closure operator: It is extensive, meaning that the convex hull of every set is a superset of . It is non-decreasing, meaning that, for every two sets and with , the convex hull of is a subset of the convex hull of . It is idempotent, meaning that for every , the convex hull of the convex hull of is the same as the convex hull of . When applied to a finite set of points, this is the closure operator of an antimatroid, the shelling antimatroid of the point set. Every antimatroid can be represented in this way by convex hulls of points in a Euclidean space of high-enough dimension. Minkowski sum The operations of constructing the convex hull and taking the Minkowski sum commute with each other, in the sense that the Minkowski sum of convex hulls of sets gives the same result as the convex hull of the Minkowski sum of the same sets. This provides a step towards the Shapley–Folkman theorem bounding the distance of a Minkowski sum from its convex hull. Projective duality The projective dual operation to constructing the convex hull of a set of points is constructing the intersection of a family of closed halfspaces that all contain the origin (or any other designated point). Special cases Finite point sets The convex hull of a finite point set forms a convex polygon when , or more generally a convex polytope in . Each extreme point of the hull is called a vertex, and (by the Krein–Milman theorem) every convex polytope is the convex hull of its vertices. It is the unique convex polytope whose vertices belong to and that encloses all of . For sets of points in general position, the convex hull is a simplicial polytope. According to the upper bound theorem, the number of faces of the convex hull of points in -dimensional Euclidean space is . In particular, in two and three dimensions the