Convex hull

Definition

Let $X$ be a subset of $\mathbb{R}^n$ . The convex hull of $X$ is the intersection of all convex sets of $\mathbb{R}^n$ that contain $X$ : $\operatorname{conv}(X) = \bigcap \{\, S \subseteq \mathbb{R}^n \;:\; S \supseteq X \text{ and } S \text{ is convex} \,\}.$

Remarks

$\operatorname{conv}(X)$ is convex, and it is the smallest convex set that contains $X$ . It is also the set of all convex combinations $\sum_{i=1}^{k} w_i x_i$ , with $k \geq 1$ , $x_i \in X$ , $w_i \geq 0$ and $\sum_i w_i = 1$ . These are the affine combinations with nonnegative weights. The convex hull of a compact set is compact, and the convex hull of an open set is open. The convex hull of a finite set is a convex polytope. For three points not on a line, it is the triangle with its inside.

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Affine combination concept
Convex set concept
Convex hull definition

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Tag	`0TCHAS04WK`
URI	`m2://0TCHAS04WK`
Kind	definition
Writing	W1 stated
MSC	52A20
Defines	Convex hull
Uses	Convex set, Affine combination

Convex hull

Definition

Remarks

Reading path

Where this leads

Further reading

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