Convex Hull

Last updated Nov 1, 2022

Let $V$ be a Vector Space over $\mathbb{R}$ and $S \subset V$. Then the Convex Hull of $S$, denoted $\mathbf{conv} S$, is defined as $$\mathbf{conv} S := \bigcap\limits {T \subset V : S \subset T, T \text{ is convex}}.$$

1. $\mathbf{conv} S$ is the smallest Convex Set containing $S$ (since Intersection of Convex Sets is Convex).
2. The Set of all Convex Combinations is the Convex Hull.
3. If $S$ is a Convex Set, then $\mathbf{conv}S = S$. This can alternatively be taken as a definition for $S$.