Convex hull - Imported Knowlpedia Content

Let $X$ be a real vector space and let $\Omega\subset X$ . The convex hull of $\Omega$ , denoted $\mathrm{co}(\Omega)$ , is defined as the intersection of all convex sets containing $\Omega$ :

\mathrm{co}(\Omega):=\bigcap\{C\subset X:\ C\text{ is convex and }\Omega\subset C\}.

Context. By stability under intersections, this definition ensures $\mathrm{co}(\Omega)$ is convex. The convex hull is the basic "convexification" operator.

Examples

If $\Omega=\{x_1,x_2\}$ , then $\mathrm{co}(\Omega)=[x_1,x_2]$ (the segment).
If $\Omega$ is the unit circle in $\mathbb{R}^2$ , then $\mathrm{co}(\Omega)$ is the closed unit disk.
If $\Omega$ is already convex, then $\mathrm{co}(\Omega)=\Omega$ .