Affine hull
# 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 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 $R^3$ is the entire space $R^3.$

## Properties

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 $X$.
* $\operatorname {aff} (S-S)=\operatorname {span} (S-S)$.
   * 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 $s_{0}\in S$, $\operatorname {aff} S=s_{0}+\operatorname {cone} (S-S)$ where $\operatorname {cone} (S-S)$ is the smallest cone containing $S-S$ (here, a set $C\subseteq X$ is a cone if $rc\in C$ for all $c\in C$ and all non-negative $r\geq 0$).
   * Hence $\operatorname {cone} (S-S)$ is always a linear subspace of $X$ parallel to $\operatorname {aff} S$.

## Related sets

* 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.
