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Convex hull
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===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>
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