Affine hull

In mathematics, the affine hull or affine span of a set S in Euclidean space Rⁿ is the smallest affine set containing S,[1] 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 aff(S) of S is the set of all affine combinations of elements of S, that is,

\operatorname {aff} (S)=\left\{\sum _{i=1}^{k}\alpha _{i}x_{i}\,{\Bigg |}\,k>0,\,x_{i}\in S,\,\alpha _{i}\in \mathbb {R} ,\,\sum _{i=1}^{k}\alpha _{i}=1\right\}.

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 R³ is the entire space R³.

For any subsets $S,T\subseteq X$

$\operatorname {aff} (\operatorname {aff} S)=\operatorname {aff} S$
$\operatorname {aff} S$ is a closed set if $X$ is finite dimensional.
$\operatorname {aff} (S+T)=\operatorname {aff} S+\operatorname {aff} T$
If $0\in S$ then $\operatorname {aff} S=\operatorname {span} S$ .
If $s_{0}\in S$ then $\operatorname {aff} (S)-s_{0}=\operatorname {span} (S-s_{0})$ is a linear subspace of $X$ .
$\operatorname {aff} (S-S)=\operatorname {span} (S-S)$ $\operatorname {aff} (S-S)=\operatorname {span} (S-S)$ .
- So in particular, $\operatorname {aff} (S-S)$ is always a vector subspace of $X$ .
If $S$ is convex then $\operatorname {aff} (S-S)=\displaystyle \bigcup _{\lambda >0}\lambda (S-S)$
For every $s_{0}\in S$ $s_{0}\in S$ , $\operatorname {aff} S=s_{0}+\operatorname {cone} (S-S)$ $\operatorname {aff} S=s_{0}+\operatorname {cone} (S-S)$ where $\operatorname {cone} (S-S)$ $\operatorname {cone} (S-S)$ is the smallest cone containing $S-S$ $S-S$ (here, a set $C\subseteq X$ $C\subseteq X$ is a cone if $rc\in C$ $rc\in C$ for all $c\in C$ $c\in C$ and all non-negative $r\geq 0$ $r \geq 0$ ).
- Hence $\operatorname {cone} (S-S)$ is always a linear subspace of $X$ parallel to $\operatorname {aff} S$ .

If instead of an affine combination one uses a convex combination, that is one requires in the formula above that all $\alpha _{i}$ 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
If however one puts no restrictions at all on the numbers $\alpha _{i}$ , instead of an affine combination one has a linear combination, and the resulting set is the linear span 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

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