Affine hull

In mathematics, the affine hull or affine span of a set $S$ in Euclidean space $ℝ^{n}$ is the smallest affine set containing $S$ , or equivalently, the intersection of all affine sets containing $S$ . Here, an affine set may be defined as the translation of a vector subspace.

The affine hull of $S$ is what $span S$ would be if the origin was moved to $S$ .

The affine hull aff( $S$ ) of $S$ is the set of all affine combinations of elements of $S$ , that is,

aff (S) = {\sum_{i = 1}^{k} α_{i} x_{i} | k > 0, x_{i} \in S, α_{i} \in ℝ, \sum_{i = 1}^{k} α_{i} = 1} .

Examples

The affine hull of the empty set is the empty set.
The affine hull of a singleton (a set made of one single element) is the singleton itself.
The affine hull of a set of two different points is the line through them.
The affine hull of a set of three points not on one line is the plane going through them.
The affine hull of a set of four points not in a plane in $ℝ^{3}$ is the entire space $ℝ^{3}$ .

For any subsets $S, T \subseteq X$

$aff (aff S) = aff S \subset span S = span aff S$ .
$aff S$ is a closed set if $X$ is finite dimensional.
$aff (S + T) = aff S + aff T$ .
$S \subset aff S$ .
If $0 \in aff S$ then $aff S = span S$ .
If $s_{0} \in aff S$ then $aff (S) - s_{0} = span (S - s_{0}) = span (S - S)$ is a linear subspace of $X$ .
aff⁡(S−S)=span⁡(S−S) if S≠∅.
- So, $aff (S - S)$ is always a vector subspace of $X$ if $S \neq \emptyset$ .
If $S$ is convex then $aff (S - S) = ⋃_{λ > 0} λ (S - S)$
For every s0∈aff⁡S, aff⁡S=s0+span⁡(S−s0)=s0+span⁡(S−S)=S+span⁡(S−S)=s0+cone⁡(S−S) where cone⁡(S−S) is the smallest cone containing S−S (here, a set C⊆X is a cone if rc∈C for all c∈C and all non-negative r≥0).
- Hence $cone (S - S) = span (S - S)$ is always a linear subspace of $X$ parallel to $aff S$ if $S \neq \emptyset$ .
- Note: $aff S = s_{0} + span (S - s_{0})$ says that if we translate $S$ so that it contains the origin, take its span, and translate it back, we get $aff S$ . Moreover, $aff S$ or $s_{0} + span (S - s_{0})$ is what $span S$ would be if the origin was at $s_{0}$ .

If instead of an affine combination one uses a convex combination, that is, one requires in the formula above that all $α_{i}$ be non-negative, one obtains the convex hull of $S$ , which cannot be larger than the affine hull of $S$ , as more restrictions are involved.
The notion of conical combination gives rise to the notion of the conical hull $cone S$ .
If however one puts no restrictions at all on the numbers $α_{i}$ , instead of an affine combination one has a linear combination, and the resulting set is the linear span $span S$ of $S$ , which contains the affine hull of $S$ .

R.J. Webster, Convexity, Oxford University Press, 1994. ISBN 0-19-853147-8.
Roman, Stephen (2008), Advanced Linear Algebra, Graduate Texts in Mathematics (Third ed.), Springer, ISBN 978-0-387-72828-5