Convex hull: Difference between revisions

Revision as of 23:23, 8 August 2024

Revision as of 23:26, 8 August 2024

Line 3:

{{good article}}

[[File:Extreme points.svg|thumb|right|The convex hull of the red set is the blue and red [[convex set]].]]

[[Intersex|In]] [[geometry]], the '''convex hull''', '''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.