Convex hull

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 $\Omega$:

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 $\mathrm{co}(\Omega)=\Omega$.