Convex hull

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Definition

Let [ilmath]S\in\mathcal{P}(\mathbb{R}^n)[/ilmath] be an an arbitrary subset of [ilmath]\mathbb{R}^n[/ilmath] - that is not empty - we define the convex hull, [ilmath]H[/ilmath] as follows:

• [math]H:\eq\bigcap_{C\in\text{Convex}(S;\mathbb{R}^n)}C[/math] - the intersection of all convex sets in [ilmath]\mathbb{R}^n[/ilmath] containing [ilmath]S[/ilmath]
  • Here [ilmath]\text{Convex}(A;\mathbb{R}^p)[/ilmath] denotes all the convex sets in [ilmath]\mathbb{R}^p[/ilmath] that contain [ilmath]A[/ilmath]

See also

• Standard [ilmath]n[/ilmath]-simplex - the convex hull of [ilmath](n+1)[/ilmath]-geometrically independent points