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Convex closure is sometimes used to refer to the convex hull, but it can also refer to the closed convex hull. See talk page for details. |
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[[File:Extreme points.svg|thumb|right|The convex hull of the red set is the blue and red [[convex set]].]]
In [[geometry]], the '''convex hull''' or '''convex envelope''' or '''convex closure'''{{refn|The terminology ''convex closure'' refers to the fact that the convex hull defines a [[closure operator]]. However, this term is also frequently used to refer to the ''closed convex hull'', with which it should not be confused — see e.g {{harvtxt|Fan|1959}}, p.48.}} of a shape is the smallest [[convex set]] that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a [[Euclidean space]], or equivalently as the set of all [[convex combination]]s of points in the subset. For a [[Bounded set|bounded]] subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around the subset.
Convex hulls of [[open set]]s are open, and convex hulls of [[compact set]]s are compact. Every compact convex set is the convex hull of its [[extreme point]]s. The convex hull operator is an example of a [[closure operator]], and every [[antimatroid]] can be represented by applying this closure operator to finite sets of points.
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==References==
{{refbegin|30em}}
*{{citation
| last = Fan | first = Ky
| title = Convex Sets and Their Applications. Summer Lectures 1959.
| url = https://books.google.com/books?id=QKkrAAAAYAAJ&pg=PA48
| publisher = Argon national laboratory
| year = 1959}}
*{{citation
| last = Andrew | first = A. M.
| |||