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

## 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 $S,T\subseteq X$
* $\operatorname {aff} (\operatorname {aff} S)=\operatorname {aff} S\subset \operatorname {span} S=\operatorname {span} \operatorname {aff} S$.
* $\operatorname {aff} S$ is a closed set if $X$ is finite dimensional.
* $\operatorname {aff} (S+T)=\operatorname {aff} S+\operatorname {aff} T$.
* $S\subset \operatorname {aff} S$.
* If $0\in \operatorname {aff} S$ then $\operatorname {aff} S=\operatorname {span} S$.
* If $s_{0}\in \operatorname {aff} S$ then $\operatorname {aff} (S)-s_{0}=\operatorname {span} (S-s_{0})=\operatorname {span} (S-S)$ is a linear subspace of $X$.
* $\operatorname {aff} (S-S)=\operatorname {span} (S-S)$ if $S\neq \varnothing$.
** So, $\operatorname {aff} (S-S)$ is always a vector subspace of $X$ if $S\neq \varnothing$.
* If $S$ is convex then $\operatorname {aff} (S-S)=\displaystyle \bigcup _{\lambda >0}\lambda (S-S)$
* For every $s_{0}\in \operatorname {aff} S$, $\operatorname {aff} S=s_{0}+\operatorname {span} (S-s_{0})=s_{0}+\operatorname {span} (S-S)=S+\operatorname {span} (S-S)=s_{0}+\operatorname {cone} (S-S)$ 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 $\operatorname {cone} (S-S)=\operatorname {span} (S-S)$ is always a linear subspace of $X$ parallel to $\operatorname {aff} S$ if $S\neq \varnothing$.
** Note: $\operatorname {aff} S=s_{0}+\operatorname {span} (S-s_{0})$ says that if we translate S so that it contains the origin, take its span, and translate it back, we get $\operatorname {aff} S$. Moreover, $\operatorname {aff} S$ or $s_{0}+\operatorname {span} (S-s_{0})$ is what $\operatorname {span} S$ would be if the origin was at $s_{0}$.

## 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 $\operatorname {cone} S$.
* 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 $\operatorname {span} S$ of S, which contains the affine hull of S.