Editing Convex hull (section)

==Topological properties==
===Closed and open hulls===
The ''closed convex hull'' of a set is the [[Closure (topology)|closure]] of the convex hull, and the ''open convex hull'' is the [[Interior (topology)|interior]] (or in some sources the [[relative interior]]) of the convex hull.{{sfnp|Sontag|1982}}

The closed convex hull of <math>X</math> is the intersection of all closed [[Half-space (geometry)|half-space]]s containing <math>X</math>.
If the convex hull of <math>X</math> is already a [[closed set]] itself (as happens, for instance, if <math>X</math> is a [[finite set]] or more generally a [[compact set]]), then it equals the closed 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.{{sfnp|Rockafellar|1970|page=99}}

If the open convex hull of a set <math>X</math> is <math>d</math>-dimensional, then every point of the hull belongs to an open convex hull of at most <math>2d</math> points of <math>X</math>. The sets of vertices of a square, regular octahedron, or higher-dimensional [[cross-polytope]] provide examples where exactly <math>2d</math> points are needed.<ref>{{harvtxt|Steinitz|1914}}; {{harvtxt|Gustin|1947}}; {{harvtxt|Bárány|Katchalski|Pach|1982}}</ref>

===Preservation of topological properties===
[[File:Versiera007.svg|thumb|The [[witch of Agnesi]]. The points on or above the red curve provide an example of a closed set whose convex hull is open (the open [[upper half-plane]]).]]
Topologically, the convex hull of an [[open set]] is always itself open, and the convex hull of a compact set is always itself compact. However, there exist closed sets for which the convex hull is not closed.<ref>{{harvtxt|Grünbaum|2003}}, p. 16; {{harvtxt|Lay|1982}}, p. 21; {{harvtxt|Sakuma|1977}}.</ref> For instance, the closed set

:<math>\left  \{ (x,y) \mathop{\bigg|} y\ge \frac{1}{1+x^2}\right\}</math>

(the set of points that lie on or above the [[witch of Agnesi]]) has the open [[upper half-plane]] as its convex hull.<ref>This example is given by {{harvtxt|Talman|1977}}, Remark 2.6.</ref>

The compactness of convex hulls of compact sets, in finite-dimensional Euclidean spaces, is generalized by the [[Krein–Smulian theorem]], according to which the closed convex hull of a weakly compact subset of a [[Banach space]] (a subset that is compact under the [[weak topology]]) is weakly compact.{{sfnp|Whitley|1986}}

===Extreme points===
{{main|Krein–Milman theorem}}
An [[extreme point]] of a convex set is a point in the set that does not lie on any open line segment between any other two points of the same set.
For a convex hull, every extreme point must be part of the given set, because otherwise it cannot be formed as a convex combination of given points.
According to the [[Krein–Milman theorem]], every compact convex set in a Euclidean space (or more generally in a [[locally convex topological vector space]]) is the convex hull of its extreme points.<ref>{{harvtxt|Krein|Milman|1940}}; {{harvtxt|Lay|1982}}, p. 43.</ref> 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.{{sfnp|Okon|2000}}