Definition
Let $S$ be a set in $\mathbb{R}^n$.
Affine hull
The affine hull of $S$ is the set of all affine combinations of elements of $S$:
\[\text{aff}(S) = \{\sum^k_{i=1} a_ix_i:k>0, x_i \in S, a_i\in \mathbb{R}, \sum^k_{i=1}a_i = 1\}.\]Convex hull
The convex hull of $S$ is the set of all convex combinations of the elements of $S$:
\[\text{conv}(S) = \{\sum^k_{i=1} a_ix_i:k>0, x_i \in S, a_i\in \mathbb{R}, a_i \geq 0, \sum^k_{i=1}a_i = 1\}.\]Difference & Examples
Difference
For the convex hull, the weights in the linear combination have an additional restriction of being non-negative. This also means that the affine hull always contains the convex hull.
Examples
- For two points in one dimension: the convex hull is the line segment joining them (including the endpoints) while the affine hull is the entire line through these two points.
- For 3 non-collinear points in two dimensions, the convex hull is the triangle with these 3 points as vertices while the affine hull is the entire plane $\mathbb{R}^2$.
References
- https://statisticaloddsandends.wordpress.com/2022/11/18/affine-hull-vs-convex-hull/