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.

Explanations

L1 human

For a finite set of points in the plane, put a rubber band round all the points and let it go. The band takes the shape of the boundary of the convex hull. The convex hull of the four corners of a square is the full square.