Affine Hull

Last updated Nov 1, 2022

Let $V$ be a Vector Space over $\mathbb{R}$ and $S \subset V$. Let $\mathcal{T} = {T \subset V : S \subset T, T \text{ is affine}}$. Then the Affine Hull of $S$, denoted $\mathbf{aff} S$, is defined as $$\mathbf{aff} S = \bigcap\limits_{T \in \mathcal{T}} T.$$

1. $\mathbf{aff} S$ is the smallest Affine Set containing $S$ (since Intersection of Affine Sets is Affine).
2. The Set of all Convex Combinations is the Convex Hull