Convex hull: Difference between revisions

Revision as of 10:40, 31 May 2025

Revision as of 01:00, 1 July 2025

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===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 (in Euclidean spaces) 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>

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(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>

TheConvex hulls can be defined more generally in certain infinite-dimensional spaces, but they may not preserve compactness in these spaces. Instead, 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===